Field Guide to Radiometry (SPIE Press Field Guide FG23)

Barbara G. Grant

Field Guide toField Guide to

SPIE PRESS | Field Guide

RadiometryRadiometry

RadiometryField Guide to

Barbara G. Grant

SPIE Field GuidesVolume FG23

John E. Greivenkamp, Series Editor

Bellingham, Washington USA

Library of Congress Cataloging-in-Publication Data

Grant, Barbara G. (Barbara Geri), 1957-Field guide to radiometry / Barbara Grant.

p. cm. – (The field guide series ; FG23)Includes bibliographical references and index.ISBN 978-0-8194-8827-51. Radiation–Measurement. I. Title.QD117.R3G73 2011535–dc23

2011033224

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Copyright © 2011 Society of Photo-Optical Instrumenta-tion Engineers (SPIE)

All rights reserved. No part of this publication may bereproduced or distributed in any form or by any meanswithout written permission of the publisher.

The content of this book reflects the work and thought ofthe author. Every effort has been made to publish reliableand accurate information herein, but the publisher is notresponsible for the validity of the information or for anyoutcomes resulting from reliance thereon. For the latestupdates about this title, please visit the book’s page on ourwebsite.

Printed in the United States of America.First printing

Introduction to the Series

Welcome to the SPIE Field Guides—a series of publica-tions written directly for the practicing engineer or sci-entist. Many textbooks and professional reference bookscover optical principles and techniques in depth. The aimof the SPIE Field Guides is to distill this information,providing readers with a handy desk or briefcase ref-erence that provides basic, essential information aboutoptical principles, techniques, or phenomena, includingdefinitions and descriptions, key equations, illustrations,application examples, design considerations, and addi-tional resources. A significant effort will be made to pro-vide a consistent notation and style between volumes inthe series.

Each SPIE Field Guide addresses a major field of opticalscience and technology. The concept of these Field Guidesis a format-intensive presentation based on figures andequations supplemented by concise explanations. In mostcases, this modular approach places a single topic on apage, and provides full coverage of that topic on that page.Highlights, insights, and rules of thumb are displayed insidebars to the main text. The appendices at the end ofeach Field Guide provide additional information such asrelated material outside the main scope of the volume,key mathematical relationships, and alternative methods.While complete in their coverage, the concise presentationmay not be appropriate for those new to the field.

The SPIE Field Guides are intended to be living doc-uments. The modular page-based presentation formatallows them to be easily updated and expanded. We areinterested in your suggestions for new Field Guide topicsas well as what material should be added to an individualvolume to make these Field Guides more useful to you.Please contact us at [email protected].

John E. Greivenkamp, Series EditorCollege of Optical SciencesThe University of Arizona

Field Guide to Radiometry

The Field Guide Series

Keep information at your fingertips with all of the titles inthe Field Guide Series:

Field Guide to

Adaptive Optics, Robert Tyson & Benjamin Frazier

Atmospheric Optics, Larry Andrews

Binoculars and Scopes, Paul Yoder, Jr. & DanielVukobratovich

Diffractive Optics, Yakov Soskind

Geometrical Optics, John Greivenkamp

Illumination, Angelo Arecchi, Tahar Messadi, & JohnKoshel

Infrared Systems, Detectors, and FPAs, Second Edition,Arnold Daniels

Interferometric Optical Testing, Eric Goodwin & JimWyant

Laser Pulse Generation, Rüdiger Paschotta

Lasers, Rüdiger Paschotta

Microscopy, Tomasz Tkaczyk

Optical Fabrication, Ray Williamson

Optical Fiber Technology, Rüdiger Paschotta

Optical Lithography, Chris Mack

Optical Thin Films, Ronald Willey

Polarization, Edward Collett

Radiometry, Barbara Grant

Special Functions for Engineers, Larry Andrews

Spectroscopy, David Ball

Visual and Ophthalmic Optics, Jim Schwiegerling


Introduction

Based on the SPIE bestseller The Art of Radiometry byJames M. Palmer and Barbara G. Grant, this Field Guideprovides a practical, hands-on approach to the subject thatthe engineer, scientist, or student can use in real time.Readers of the earlier work will recognize similar topicsin condensed form, along with many new figures and achapter on photometry.

Written from a systems engineering perspective, thisbook covers topics in optical radiation propagation, ma-terial properties, sources, detectors, system components,measurement, calibration, and photometry. Appendicesprovide material on SI units, conversion factors, sourceluminance data, and many other subjects. The book’s orga-nization and extensive collection of diagrams, tables, andgraphs will enable the reader to efficiently identify andapply relevant information to radiometric problems aris-ing amid the demands of today’s fast-paced technical envi-ronment.

I gratefully acknowledge the contributions to my educa-tion and career from three professors of Optical Sciencesat the University of Arizona, gentlemen all. They are thelate Jim Palmer (1937–2007), who mentored me in radiom-etry for many years and provided me the opportunity tocomplete The Art of Radiometry; Emeritus Professor PhilSlater, who selected me as a graduate student and trainedme in remote sensing, and who continues to encourageand support me; and Eustace Dereniak, who generouslyshared his knowledge from the very start, provided me myfirst opportunities to teach, and has strongly supportedmy career for more than twenty years. To all, my heart-felt thanks.

This book is dedicated to my family and particularly to thememory of my father, William Grant of Chicago, Illinois,a US Navy veteran of WWII who taught me to play the“Garryowen” as soon as my fingers could reach a pianokeyboard.

Barbara G. GrantSeptember 2011


vii

Table of Contents

Glossary of Symbols and Notation xi

Introduction to Radiometry 1The Electromagnetic Spectrum 1The Basics 2

Propagation of Optical Radiation 3Plane and Solid Angles 3Projected Area and Projected Solid Angle 4f /# and Numerical Aperture 5Radiometric Quantities Summarized 6Photon Quantities 7Spectral Radiant Quantities 8Radiance, Radiant Exitance, and Irradiance 9Exitance–Radiance Relationship 10Intensity 11Isotropic and Lambertian Sources 12Inverse Square Law of Irradiance 13Cosine3 and Cosine4 Laws of Irradiance 14Throughput and Its Invariance 15Area and Solid Angle Products 16Basic Radiance and Radiance Invariance 17The Equation of Radiative Transfer 18Configuration Factors 19Power Transfer: Point Source 20Power Transfer: Extended Source 21Power Transfer: Field Lens Added 22Irradiance from a Lambertian Disk 23Irradiance from a Lambertian Sphere 24The Integrating Sphere 25Camera Equation and Image Plane Irradiance 26

Radiometric Properties of Materials 27Overview of Material Properties 27Transmission 28Reflection 29Absorption and the Conservation of Energy 30Emission 31Specular Transmissivity and Reflectivity 32Single-Surface Illustrations 33More on Specular Propagation 34


viii

Table of Contents

Transmission: Absorbing and Reflecting Materials 35Materials as Targets 36Optical Material Selection Considerations 37

Generation of Optical Radiation 38Planck’s Law 38Stefan–Boltzmann and Wien Displacement Laws 39Rayleigh–Jeans Law and Wien Approximation 40Radiation Laws in Terms of Photons 41Kirchoff ’s Law 42Natural Sources 43Lambert–Bouguer–Beer Law and Langley Plot 44Artificial Sources 45Luminescent Mechanisms 46Some Luminescent Sources 47

Detectors of Optical Radiation 48Detector Types 48Detector Definitions 49More Detector Definitions 50Detector Figures of Merit 51Noise Concepts and Definitions 52The Most Unpleasant Noises 53More Unpleasant Noises 54Thermal Detectors 55Thermoelectric Detectors 56The Bolometer 57Pyroelectric Detectors 58Photon Detectors 59Photoconductive Detectors 60Photoemissive Detectors 61Photovoltaic Detectors 62Photovoltaic Current and Performance 63Detector Interfacing 64Single and Multiple Detectors 65Detector Array Architectures 66Choosing a Detector 67

Radiometric System Components 68Choppers and Radiation References 68Baffles and Cosine Correctors 69Spectral Separation Mechanisms 70


ix

Table of Contents

Prisms and Gratings 71Filters 72

Calibration and Measurement 73Radiometric Calibration Basics 73Radiometric Calibration Philosophy 74Distant Small Source Calibration 75Collimators and the Distant Small Source 76More on Collimators 77Extended Source Calibrations 78Other Calibration Methods 79The Measurement Equation 80Errors in Measurements 81Signal-to-Noise Ratio and Measurement Error 82The Range Equation 83Radiometric Temperatures 84

Photometry 85Photometric Quantities 85Human Visual Response 86Color 87Sources and the Eye’s Response 88

Appendices 89SI Base Quantities, Prefixes, and Uncertainty

Reporting 89Physical Constants: 2010 CODATA Recommended

Values 90Source Luminance Values 91More Source Values 92Solid Angle Relationships 93Rays, Stops, and Pupils 94Diffraction 95Action Spectra and Optical Radiation Regions 96

Equation Summary 97

References 109

Bibliography 110

Index 112


xi

Glossary of Symbols and Notation

Ap Projected areaAs Source areaAS Aperture stopAsph Sphere surface areaB Effective noise bandwidthBLIP Background-limited infrared photodetectorBRDF Bidirectional reflectance distribution functionBTDF Bidirectional transmittance distribution

functionc Velocity of optical radiation in vacuumC CoulombC Capacitancec1 First radiation constantc2 Second radiation constantCCD Charge-coupled deviceCID Charge-injection deviceCMOS Complementary metal-oxide semiconductorCTE Charge-transfer efficiencyd DistanceD DetectivityD Entrance pupil diameterD∗ Specific detectivityD∗∗ Specific normalized detectivityD∗

q Specific detectivity in photonsDQE Detective quantum efficiencyE IrradianceE f Fermi levelEg Gap energyEq Photon irradianceEv Illuminancef Focal lengthf Ratio of port area to integrating sphere areaF Configuration factorF Noise factorf /# F-numberf /#′ Working f /#fc Cutoff frequencyFc Chopping factor


xii


FOV Field of viewFS Field stopfT Thermal cutoff frequencyG Photoconductive gainG Radiant power gainh Planck’s constantH Heat capacityH Radiant exposureHq Photon exposureHv Photometric exposureI Radiant intensityi Average photocurrent due to mean carrier

numberIdc Direct currentIFOV Instantaneous field of viewin Noise currentIq Photon intensityis Signal currentIv Luminous intensityj

p−1k Boltzmann’s constantK Thermal conductanceKm Luminous efficacy (photopic)K ′

m Luminous efficacy (scotopic)Ks 1/ f noise source-dependent constantl Source maximum linear dimensionL RadianceLq Photon radianceLSB Least significant bitLv Luminancem Airmassm Grating order numberm MagnificationM Radiant exitancemp Pupil magnificationMq Photon exitanceMv Luminous exitancen Number of bits in a digital wordn Refractive index


xiii


n Surface normal vectorN Mean number of carriersN Number of illuminated grating groovesn Average number of photonsNA Numerical apertureND Neutral densityNEI Noise equivalent irradianceNEP Noise equivalent powerNF Noise figureNP Entrance pupilp Pyroelectric coefficientPs Spontaneous polarizationq Electronic chargeQ,Qe Radiant energyQp,Qq Photon energyQrp Resolving powerQv Luminous energyr Radius of circle or sphereR RangeR Reflectance factorR ResistanceR ResponsivityRA Resistance-area productRE Irradiance responsivityRL Detector load resistanceRL Radiance responsivityRq Photon responsivityRT Thermal resistanceRΦ Power responsivityS Seebeck coefficientSNR Signal-to-noise ratiot TimeT Absolute temperatureT ThroughputTc Curie temperatureTn Noise temperaturev Velocity of optical radiation in a mediumVB Detector bias voltagev j Johnson noise voltage


xiv


vn Noise voltagevrms Root-mean-square voltageVs,vs Signal voltageV (λ) Photopic visual responseV ′(λ) Scotopic visual responsex Path lengthXP Exit pupilX ,Y , Z Tristimulus values based on the 1931 CIE

Standard ObserverZ Impedanceα Absorptanceα 1/ f noise expression constantα′(λ) Spectral absorption coefficientβ 1/ f noise expression constantβ Temperature coefficient of resistance (TCR)ε Emittanceη Responsive quantum efficiencyθ Angle of incidence, reflection, refractionθ Angle from normal incidence or emittanceΘ1/2 Half-angle of illumination coneκ(λ) Spectral extinction coefficientλ Wavelengthλ0 Wavelength in vacuumλc Cutoff wavelengthµ Carrier mobilityΠ Peltier coeffficientρ Reflectanceρp Polarized reflectivity parallel componentρs Polarized reflectivity perpendicular

componentρss Single-surface reflectanceσ Standard deviationσ Stefan–Boltzmann constantσ2 Varianceσe Electrical conductivityτ Time constantτ Transmittanceτ0(λ) Spectral optical thicknessτatm Atmospheric transmittance


xv


τi Internal transmittanceτl Carrier lifetimeτlens Lens transmittanceτo Optical transmittanceτp Polarized transmissivity parallel componentτs Polarized transmissivity perpendicular

componentτss Single-surface transmittanceτT Thermal time constantφ Rotational angleφ Work function in a photoemissive detectorφ0 Potential barrier height in a photovoltaic

detectorΦ Radiant powerΦq Photon fluxΦv Luminous powerω Radian frequency (2π f )ω Solid angleΩ Projected solid angleΩo Unit projected solid angle


Introduction to Radiometry 1

The Electromagnetic Spectrum

The practical electromagnetic spectrum extends from dcto frequencies greater than 1020 Hz. We shall considerthe optical portion of the spectrum to cover the five-decade (frequency) range from 3 × 1011 to 3 × 1016 Hz,or roughly 10 nm to 1000 µm (1 mm), and encompassesthe ultraviolet, visible, and infrared regions. At shorterwavelengths are x rays, gamma rays, and cosmic rays; atlonger wavelengths are microwaves and radio waves.

UVC, UVB, UVA 200 nm–400 nmVisible 360 nm–780 nmNear IR 0.78 µm–1.1 µmShortwave IR (SWIR) 1.1 µm–2.5 µmMidwave IR (MWIR) 2.5 µm–7 µmLongwave IR (LWIR) 7 µm–15 µm

Wavelength regions and subdivisions of the spectrumoften overlap and are not always consistent from oneauthority to another.

Radiation from the far UV through x rays and higherfrequencies is called ionizing radiation, as it allows anelectron to be separated from an atom or molecule.


2 Introduction to Radiometry

The Basics

Radiometry is the measurement of optical radiantenergy. It also refers more generally to the principles andlaws behind the generation, propagation, and detection ofoptical radiation.

The velocity of light c in vacuum is a constant equal toabout 3.00×108 ms−1, a faster rate of travel than in anyother medium. A medium’s refractive index n is the ratioof c to the velocity of propagation v in that medium,n = c/v.

Snell’s law relates refractive index n to angle of propaga-tion θ within adjacent media due to boundary transition(refraction):

n1 sinθ1 = n2 sinθ2.

When n2 > n1, the refracted raytravels closer to the boundarynormal than does the incidentray. This makes intuitive sensebecause the sine of a smallangle is less than the sine ofa larger one when 0- to 90-degangles are considered.

The wavelength of a monochromatic source in a vacuumis related to its wavelength in any medium by λ0 = nλ.

If radiation is outside the visible spectrum, you cannotcall it “light.” The visible spectrum extends from about380 to 760 nm and is perceivable by the normal humaneye. Infrared and ultraviolet optical radiation propagatebeyond these limits; therefore, there is no such thing as“infrared light” or “ultraviolet light.”


Propagation of Optical Radiation 3

Plane and Solid Angles

Plane angle θ is theangle formed by tworadii of a circle. It isdefined as the ratioof the arc length tothe circle’s radius, θ =l/r. Its unit is theradian, the plane anglebetween two radii of acircle that cut off, onthe circumference, anarc equal in length to the radius. There are 2π radians(360 deg) in a circle.

Solid angleω is the ratio of a subtended spherical surfacearea to the square of the sphere radius. Its unit is thesteradian (sr), the solid angle with vertex at the center ofa sphere, subtending an area on the surface of the spherewhose sides have length equal to the sphere radius. Thesolid angle of a hemisphere is 2π sr, and the solid angle ofa sphere is 4π sr.

dω= dAsph

r2 = sinθdθdφ

ω=∫φ

∫θ

sinθdθdφ

For a right circular cone,ω = 2π(1 − cosΘ1/2), whereΘ1/2 is the cone half-angle.


4 Propagation of Optical Radiation

Projected Area and Projected Solid Angle

Projected area Ap is the recti-linear projection of a surface ofany shape onto a plane. In dif-ferential form, its expression isdAp = cosθdA, where θ is theangle between the line of obser-vation and the surface normal.

Integrating over the surface area, the expression becomesAp = ∫

A cosθdA.

Shape Surface area Projected area

Rectangle A = LW Ap = LW cosθ

Flat circle A =πr2 =πd2/4 Ap =πr2 cosθ

Sphere A = 4πr2 =πd2 Ap = A/4=πr2

Projected solid angle Ω is the solid angle ω projectedonto the plane of the observer. It may be visualized asthe projection of dω onto the base of the unit hemisphere.It incorporates an additional cosine term over that in thedefinition of solid angle.

The projected solid angle of a hemisphere is π sr, and itsunit is the steradian.

dΩ= cosθdω= cosθsinθdθdφ

For a right circularcone,

Ω=∫ 2π

0dφ

∫ Θ1/2

0sinθcosθdθ

=πsin2Θ1/2



f /# and Numerical Aperture

The f /# of an optical system is most often seen defined asthe ratio of the effective focal length (object at infinity) ofthe system to the diameter D of its entrance pupil:

f /#= fD

Good optical systems ful-fill the Abbe sine con-dition, with a sphericalwavefront converging tothe focal point, and thepreferred definition is

f /#= 12sinΘ1/2

This definition implies that Ω = π / [4( f /#)2]. The two ex-pressions for f /# are equivalent only paraxially, that is,for small values of Θ1/2 such that sinΘ1/2 = tanΘ1/2 =Θ1/2.

In cases where image or object are not at infinity, aworking f /#, f /#′, may be defined as

f /#′ = ( f /#)[1+ (m/mp)]

where mp (the pupil magnification) is 1 for a singlelens or mirror, and magnification m is the ratio of imageheight to object height.

Numerical aperture (NA) is the sine of the half-angle ofthe largest axial ray cone that can enter or leave a system,multiplied by the refractive index of the medium in whichthe cone’s vertex is located: N A = nsinΘ1/2.

Optical designers refer to the f /# as a measure of thesystem’s speed, but not in the sense of a mechanicalscan rate. “Fast” systems gather light more quickly than“slow” systems do; the former have lower f /#s than thelatter.



Radiometric Quantities Summarized

Quantity Symbol Definition Units

Radiant energy Q or Qe — joule (J)

Radiant power(flux)

ΦdQdt

watt (W)

Radiant intensity IdΦdω

W/sr

Radiant exitance MdΦdA

W/m2

Irradiance EdΦdA

W/m2

Radiance Ld2Φ

dAdωcosθW/m2sr

The above derived quantities originate from SI basequantities and are so called by the Bureau Internationaledes Poids et Mesures and other standards organizations.

Radiant exposure H is defined by NIST as dQ/dA, thederivative of radiant energy with respect to area. It hasunits of J/m2 and is featured in biological applications oflight such as phototherapy. It is also defined as

∫Edt, the

integral of irradiance over time. Analogous expressionsexist for photon and photometric exposure.



Photon Quantities

Photon quantities are defined analogously to radiomet-ric quantities. They carry the same symbols as radiometricquantities and are subscripted with p or q.


Photon energy Qq # photons —

Photon flux ΦqdQq

dt/s

Photon intensity IqdΦq

dω/s sr

Photon exitance MqdΦq

dA/s m2

Photon irradiance EqdΦq

dA/s m2

Photon radiance Lqd2Φq

dAdωcosθ/s m2sr

For easy conversion from W to photons/s at wavelength λ0,use Φq = Φ×λ0/hc, where h is Planck’s constant and c isthe speed of light in a vacuum.

At λ = 1 µm, 5.034×1018 photons/s = 1 W; and 1 photonhas energy Q = 1.986×10−19 J. For energy in electron volts(eV), divide Q by the electronic charge q, 1.602×10−19 C,to obtain 1.239 eV.

If broadband, integrations must be performed:

Φq = 1hc

∫ ∞

0λΦλdλ and Φ= hc

∫ ∞

0

(Φλ,qdλ

λ

)



Spectral Radiant Quantities

Spectral radiant quantities are defined similarly totheir radiometric counterparts and include a derivativewith respect to wavelength (or frequency), as thoughan instrument were able to record the value in aninfinitesimally narrow region. This is indicated by thesubscript λ. In the table below, the wavelength unit usedis meters (m), but the definitions are equally valid forother wavelength units such as nanometers (nm) andmicrometers (µm).


Spectral radiantenergy

QλdQdλ

J/m

Spectral radiantpower

ΦλdQλdt

= dΦdλ

W/m

Spectral radiantintensity

Iλd2Φ

dωdλW/sr m

Spectral radiantexitance

Mλd2Φ

dAdλW/m2m

Spectralirradiance

Eλd2Φ

dAdλW/m2m

Spectralradiance

Lλd3Φ

dAdωcosθdλW/m2sr m

The wavelength distribution of a spectral radiant quan-tity is described as a function of λ, and it is common tosee an integration performed over a wavelength band,symbolized as L∆λ =

∫∆λLλdλ.



Radiance, Radiant Exitance, and Irradiance

Radiance L [W/m2 sr] is the elemental quantity of ra-diometry, expressing the power per unit area and per unitprojected solid angle from a source; or alternatively, thepower per unit projectedarea per unit solid angle.Radiance is directional innature, and the sourcesfrom which it emanatesmay be active (a thermalor luminescent emitter) orpassive (reflective). As afield quantity, it can be evaluated at any point along abeam in the direction of its propagation.

Radiant exitance M [W/m2]is the power per unit arearadiated into a hemisphere;accordingly, it is a measure offlux density.

Mathematically, it may beobtained from radiance by

integrating over the projected solid angle of a hemisphere,π sr: M = ∫

πLdΩ.

Irradiance E [W/m2] isthe opposite of exitance,as it is the power perunit area from a hemi-sphere incident upon asurface. Irradiance maybe obtained from radianceby integrating over theprojected solid angle of ahemisphere, π sr: E = ∫

πLdΩ.



Exitance–Radiance Relationship

The relationship betweenradiant exitance and ra-diance is, in general, com-plex, and depends greatlyon the angular distribu-tion of radiance from thesource, L(θ,φ).

The power transferred fromdA1 (the source) to dA2 (onthe surface of the sphere) is

d2Φ= L(θ,φ)dA1 cosθdA2

r2

where θ is the angle between the surface normal to dA1and the direction of propagation. By definition,

dA2 = dω× r2

Substituting for dω in terms of θ and φ,

dA2 = r2 sinθdθdφ

Then, the power transferred to the surface dA2 is

d2Φ= L(θ,φ)dA1 sinθcosθdθdφ

By applying limits for a hemisphere to θ and φ, andconstructing an integral,

Φ=∫ 2π

0dφ

∫ π/2

0L(θ,φ)A1 sinθcosθdθ

If the source at A1 is Lambertian, then

Φ= 2πLA1

(sin2θ

2

)π/2

0

Substituting M =Φ/A1 and solving produces M = πL; thatis, the radiance and radiant exitance of a Lambertiansurface are related by a factor of π—the projected solidangle of a hemisphere.



Intensity

Radiant intensity I[W/sr] is the power perunit solid angle emittedfrom a source. It is as-sociated with a pointsource, one that is smallrelative to the distance

between the source and target or measurement location. Itmay be obtained from radiance by integrating over area:I = ∫

A LdA. Radiant power may be obtained from inten-sity by integrating over solid angle ω: Φ= ∫

ω Idω. For anisotropic point source, the power Φ is 4π, due to the factthat the source radiates uniformly into a sphere of 4π sr.

The intensity of a distant star may be determined if wehave measurements of its irradiance and distance (thelatter can be provided by interferometric techniques).

Intensity is the most frequently misused term inradiometry, used incorrectly to represent quantitiesincluding flux density and radiance. “Intensity” and“brightness” are often synonyms in common Englishusage, but not in radiometry. To avoid confusion, note that(luminous) intensity is an SI base quantity; any referenceto the term other than power per unit solid angle isincorrect according to SI terminology.

A point source does not exist in reality; if it did, it wouldhave infinite energy density, which is not physicallypossible. The mathematical construct of a point sourcesimplifies calculations in many circumstances. Yet in thiscase, as in many others, it is important to make surethat the underlying assumption holds: the source mustbe suitably distant from the receiver or observation point.



Isotropic and Lambertian Sources

An isotropic point sourcehas the same intensity inall directions; this implies aspherical source. A distantstar may be considered to ap-proximate an isotropic pointsource.

A Lambertian source is one whoseradiance is independent of direction:L(θ,φ) = Constant. The term refers toa flat surface or to a flat element of anonflat surface.

When viewed from a distance, the source radiance and itsarea projected in the direction of observation comprise itsintensity:

I = LAp

This intensity falls off as the cosine of the angle betweenthe surface normal and observation direction,

I = I0 cosθ

which is Lambert’s law. A good example of a Lambertiansurface is one covered by matte white paint. Underuniform illumination, it appears equally bright from allobservation directions.

The Lambertian approximation applies to many differenttypes of surfaces, even to polished metals far from theangle of specular reflection. It is best to confirm thisapproximation with measurement whenever possibleand keep track of its effects on radiometric calculations.



Inverse Square Law of Irradiance

The inverse square law of irradiance, one of the mostwidely used approximations in radiometry, states that theirradiance from an isotropic point source varies inverselywith the square of the source distance. In its most generalform, it includes the angle between the normal to thereceiving (target) surface and the source–receiver axis:

E = I cosθd2

A dimensional analysis of the units on the right-hand sideof the equation appears to reveal a discrepancy: while theunits of irradiance are W/m2, the intensity term providesa per steradian unit as well. A “unit solid angle,” Ωo =1 sr, has been proposed, leading to E = IΩo cosθ/d2, butthe Ωo term is likely to be confused with a free variablerather than being used as a constant. Instead, it should beremembered that plane and solid angles, being ratios, aredimensionless.

The inverse square law works well for small objects andlarge distances. Let l be the source’s largest dimension(diameter or length):

• For l < d/10 (2.9 deg half-angle), the error in applyingthe law is <1%.

• For l < d/20 (1.4 deg half-angle), the error in applyingthe law is <0.1%.

If the conditions to apply the inverse square law do nothold, the sourcemust be consid-ered in terms ofits area and ra-diance.



Cosine3 and Cosine4 Laws of Irradiance

For a target at P oriented toward a point sourcewith intensity I, the irradiance is described bythe inverse square law, E = I/d2. If the point P′ isdrawn a linear distance from P, the irradiance on atarget at P′ perpendicular to the source is

E = Is

D2 = Is cos2θ

d2

If the target is now rotated sothat it is parallel to segmentPP′, another cosine term isapplied, and the irradiance atpoint P′ is

E = Is cos3θ

d2

This is the cosine3 law of irra-diance, valid for an isotropic pointsource and similar geometry.

If the point source is replaced by anextended, Lambertian source ori-ented normal to P, the irradianceat P′ must include another cosineterm to account for source projec-tion. The extended source must becharacterized in terms of its areaand radiance as

E = LAs cos4θ

d2

which is the cosine4 law ofirradiance.



Throughput and Its Invariance

Throughput T, also called étendue, geometrical extent,and AΩ product, is the product of the cross-sectionalarea of a beam and the projected solid angle whose apexis at the areal cross-section and that subtends the nextarea in the chain. Angles are subscripted so that the firstnumber is the area subtended and the second numberis the location of the apex. Therefore, ω21 is the solidangle subtended by surface 2 at surface 1. Throughput isinvariant within an optical system, T = AΩ=Constant.

Basic throughput is analogously defined, with theinclusion of the refractive indices of two media on oppositesides of a lossless boundary, n2T = n2 AΩ=Constant.

The optimal case for throughput analysis arises whenthe source image entirely fills the detector, there are notransmission losses, and all surfaces are on axis. For thatcase,

AsΩos = AoΩso = AoΩdo = AdΩod

In cases where the paraxial approximation does not apply,a lens must still satisfy the Abbe sine condition for theabove relationships to hold.



Area and Solid Angle Products

In any analysis of throughput within an optical system,it is important to use the correct area–solid angle pair.That pair will always include theangle having its apex at the areaunder consideration. The maxim“No ice cream cones!” should beapplied.

When the source image is largerthan the detector (extendedsource), do not use AsΩos orAoΩso, as the radiance from theentire source area will not trans-fer through the system to reachthe detector.

When the source image is smaller than the detector(point source, which may be obtained when viewing theLambertian area As from sufficient distance), do not useAdΩod or AoΩdo, as the detector area has little meaningwhen not filled with the source image.



Basic Radiance and Radiance Invariance

If rays are tracedalong a losslessboundary betweentwo media havingdifferent indices ofrefraction, the solid

angle changes according to Snell’s law. Taking this changeinto account and applying the principle of conservation ofenergy, the quantity L/n2 is invariant across the boundary.This quantity is called basic radiance and is used whenthe object and image are located in media with differentrefractive indices.

Radiance was previously defined as power from a sourceper unit area and per unit projected solid angle, oralternatively, as power from a source per unit projectedarea per unit solid angle. The mathematical expressionscorresponding to the two cases are

L = d2Φ

dAdΩand L = d2Φ

dApdω

In the absence of sources or sinks along a beam path,and neglecting losses including those due to transmission,reflection, and scattering, power within the beam isconserved. Since throughput is conserved (invariant) in anoptical system, radiance must also be invariant in orderfor conservation of power to be obeyed.

The results of this radiance invariance include:

(1) The radiance of an image at the detector plane of acamera is the same as the radiance of the scene, assumingno transmission losses due to atmosphere and optics; and

(2) The radiance at the focal plane of a radiometer is thesame as that of the target if there are no transmissionlosses due to atmosphere and optics.

These results greatly simplify radiometric calculations.



The Equation of Radiative Transfer

The equation of radiative transfer is arguably the mostimportant equation in radiometry, as it provides the math-ematical description for transfer of power (flux) betweentwo surfaces. In differential form, it is

d2Φ1→2 = L(θ,φ)dA1 cosθ1dA2 cosθ2

d2

where L is the radiance at surface 1.

In integral form, the equation is

Φ1→2 =∫

A2

∫A1

L(θ,φ)cosθ1 cosθ2

d2 dA1dA2

In order to integrate (sum) the incremental contributionsto radiant power, the source must be incoherent (beamcomponents not in phase) and there must be no interfer-ence effects in the beam.

If the following conditions hold, the equation may be sim-plified:

• Lambertian source so that L(θ,φ) is a constant;• Distance d is much larger than the maximum linear

dimension (diameter or length) of A1 and A2; and• Both areas are on axis so that θ1 and θ2 are zero and

their cosines are unity.

If all conditions are met, then

Φ1→2 = LA1Ω21 = LA2Ω12 = LAΩ



Configuration Factors

Used extensively in heat transfer cal-culations, configuration factors ex-press the geometrical relationships un-derlying power transfer between twosurfaces. The transfer may occur be-tween two differential elements of area,between a differential element and afinite area, or between two finite ar-eas. The major advantage of configu-ration factors is the existence of manysolved geometries easily accessible in

print and on-line catalogues. Configuration factor F is de-fined generally as

F = Φ1→2

Φ1

where Φ1→2 is the power reaching surface 2 from surface 1,and Φ1 is the power leaving surface 1. It is essentialthat the source surface be Lambertian; its radiance isLs. For the case of transfer between two differential areaelements, the expression becomes

dFd1,d2 =cosθ1 cosθ2

πd2 dA2 = 1π

cosθ1dΩ21

where Ω21 is the projected solid angle subtended bysurface 2 at surface 1, and the θi are the angles betweenthe surface normals and the axis of distance d connectingthe two surfaces. The configuration factor relating adifferential element and a finite area is

Fd1,2 =∫

A2

cosθ1 cosθ2

πd2 dA2 =∫

A2

dFd1,d2

If both areas are finite, then

F12 = 1A1

∫A1

∫A2

cosθ1 cosθ2

πd2 dA2dA1 and

Φ1→2 = Ls A1Ω21 =πLs A1F12 = M1 A1F12



Power Transfer: Point Source

In the case of a two-aper-ture radiometer, with thedetector active area de-fined by aperture area A2,and a point source atinfinity providing a colli-mated beam, the poweron the detector is given by

Φd = Ed A2, where Ed = Is

(D+S)2

according to the inverse square law of irradiance.

Adding a lens at A1 causes the beam to focus on thedetector, and the power transferred now becomes

Φlens = Elens Alens =Is Alens

D2 , with Φd = τlensΦlens

Assuming that the distance D is large compared to S, thegain in power using a lens is simply the ratio of the areastimes the lens transmittance

G = τlens Alens

A2

For maximum power transfer in this configuration, makeτlens and the area ratio as large as possible, takingcare that the image remains fully contained (that is,unvignetted) within A2.



Power Transfer: Extended Source

For an extended source,we use the simplificationof the equation of radia-tive transfer, Φd = Ls AΩ.Because the source is ex-tended, we do not use itsarea in computation, asthat would have no mean-ing. At area A2, Ω12 is the solid angle subtended by areaA1 (notional surface). The equation for power transferredto the detector is

Φd = Ls A2Ω12 = Ls A1 A2

S2

(Note that the same result could be obtained by choosingA1 as the area.)

The area–solid angle rela-tionship is more preciselyillustrated with a lens inplace. Again using A2 andΩ12, the power on the de-tector is

Φd = τlensLs A2Ω12 = τlensLs A1 A2

S2

In other words, by placing a lens in the system withan extended source, we have reduced the power on thedetector by the transmission of the lens.

The aperture that limits the cone of light entering asystem from an on-axis point is called the aperturestop. This is A2 in the case with no lens, and A1 witha lens. The field stop limits the extent the detector cansee; it is A1 in the case with no lens, and A2 with a lens.



Power Transfer: Field Lens Added

Improvement to the quality of the image may be made byadding a field lens to image the aperture stop onto thedetector. This approach has several advantages over thelens/detector combination, including uniform irradiancedistribution over the detector and the ability to adjust theposition of the field lens without changing the position ofthe detector (the latter being difficult to do, in practice,as the detector assembly includes electronics and otherwiring).

In this case, theimage of the aper-ture stop typicallyunderfills or over-fills the detector.

Φd = Ls AoτoΩ f s−o =τoLs Ao A f s

f 2

If it underfills, all power transferred thus far will be trans-ferred to the detector. Optical transmission τo is the prod-uct of both lens’ transmittances.

If it overfills, then

Φd = τoLs AdΩ f sd = τoLs Ad A f s

d2

where d is the distance from the detector to the field stop.If the image of the aperture exactly fills the detector, eitherexpression for power transfer may be used.

A field lens helps control stray radiation within thesystem. A chopper modulating the input signal may beplaced at the field stop and not directly in front of thedetector. Multiple detectors sharing a common aperturemay be used.



Irradiance from a Lambertian Disk

A common radia-tive transfer config-uration involves adisk and a detectorplaced on axis sothat no cosine termsare needed to de-scribe a projected area. If the disk is Lambertian, thenthe simplification of the equation of radiative transfer canbe applied:

Φs→d = Ls AdΩsd

The irradiance at the detector is

Ed = Φd

Ad= LsΩsd = Lsπsin2Θ1/2 = Ls

π

4( f /#)2

Expressing relevant quantities in terms of the source-detector distance and source area, we have

sin2Θ1/2 = a2

a2 +b2 and Ed =πLsa2

a2 +b2

The latter result allows us to calculate the irradiance atthe detector for a variety of source–detector distances aswe change distance b. This is particularly significant whendeciding where to place a source (or detector) for accurateradiometric calibration.

Distance b Half-angle Θ1/2 Irradiance Edb À 2a Very small πL (a2/b2), Is/b2

b = 2a 26.5 πL/5b = a 45 πL/2b = 0 90 πL

If the disk is not Lambertian, radiance is not independentof direction, and integration is required to obtain its value.Closed-form solutions are rare, and numerical integrationmethods are typically employed for this task.



Irradiance from a Lambertian Sphere

Case I: Sphere surface at distance b from detector:

Ed =πLa2/(a2 +2ab+b2) sin2Θ1/2 = a2/(a+b)2


b = 2a 19.5 πL/9b = a 30 πL/4b = 0 90 πL

Case II: Sphere center at distance b from detector:

Ed =πLa2/b2 = Ls Ap/b2 = Is/b2 sin2Θ1/2 = a2

b2


b = 2a 30 πL/4b = a 90 πL

Notes: (1) At b À 2a, the source’s shape doesn’t matter,as it is considered a point source. (2) Case II shows thatthe inverse square law holds for any Lambertian sphereand distance when distance is measured from the sphere’scenter—mathematically true, but important to measure.



The Integrating Sphere

An integrating sphere providesa spatially uniform source of ra-diance due to multiple reflectionsfrom its interior coating (Lamber-tian). Because r is the sphere ra-dius, θ1 = θ2 = θ; cosθ = d/2r. As-sume that dA1 = dA2 = dA. Em-ploying the equation of radiativetransfer,

d2Φ1→2 = L dA 2 cos2θ

d2 = L dA 2 d2

4r2d2 and dE = LdA/4r2

For a Lambertian surface, L = Eρ/π, so dE = EρdA/4πr2.This is the irradiance on a small element of area dA.Integration over the interior surface is complex and musttake account of multiple reflections. The bottom line is

E = ρΦ

4πr2(1−ρ)

where Φ is the power on a source placed into the spherethrough a port. This irradiance—constant over the sphereby virtue of geometry (dependent only on r) and varyingonly with small variations in reflectance ρ—makes thesphere a useful radiance source.

Real spheres are fitted with ports and baffles, havenonuniform coatings, and are never exactly Lambertian.The radiance in a practical sphere may be estimated asL = ρΦ/πAsph[1−ρ(1− f )], where f is the ratio of the totalport area to that of the sphere. Uses for these spheresinclude

• Uniform radiance source• Depolarization• Light source mixing; color mixing• Cosine-corrected receivers• Reflectance and transmittance measurement



Camera Equation and Image Plane Irradiance

The camera equation describes image irradiance at thefocal plane for a system operating at finite conjugates; mis the magnification.

E t = πLs

4( f /#)2(1+m)2

For an object at infinity, m = 0 and E t = πLs/4( f /#)2. Forequal conjugates, m = 1 and E t =πLs/16( f /#)2.

More general equations for image plane irradiance,from extended and point sources respectively, are

E t = τoτatmπ fv(1− A2)Ls(θ,φ)cosnθ

4( f /#)2(1+m)2and

E t = τoτatm fv(1− A2)Is cosnθ

d2

where τo is optical transmittance, τatm is atmospherictransmittance, fv is a vignetting factor, and A = (obscura-tion diameter/primary mirror diameter) for centrally ob-scured systems. The variable n accounts for projections ofsource and target areas, as discussed previously. It is 3 foran isotropic (point) source and typically 4 for an extendedLambertian source.

Why would the variable n be typically 4, rather thanalways 4, for an extended Lambertian target given thecosine4 law? The answer is that good optical designershave been able to reduce n to 3 by deliberately introduc-ing aberrations into certain optical designs.1 Note thatthis also points to the importance of specifying exactlywhere irradiance is being calculated. It is best not to“plug and chug!”


Radiometric Properties of Materials 27

Overview of Material Properties

Optical radiation incident upon a boundary betweenmedia may be reflected, transmitted, or absorbed,with the process or processes occurring dependent on thewavelength and directional properties of the incomingradiation, and the properties of the material.

Reflectance, transmittance, and absorptance are all ratiosof output to input radiometric quantity. This makes themdifferent from radiant power, radiance, irradiance, etc.,which are defined in terms of derivatives. The distinctionis important, particularly when measurements over asmall wavelength region are considered. While the radiantpower at a particular wavelength is Φλ, the subscriptdenoting a spectral region as small as the measurementequipment makes possible, the material property P(λ) atthat wavelength is P(λ) = Φλo/Φλi, the subscripts o andi denoting output and input, respectively. Transmittance,reflectance, and absorptance are never actually measured,only calculated.

Considerable debate exists over the suffixes “-ance” and“-ivity” in material properties definition. The usage herereserves terms ending in “-ivity” for the properties of apure substance (e.g., the reflectivity of pure aluminum,calculated from its complex index of refraction with nand κ vs. the reflectance of a particular specimen of 6061aluminum with surface structure associated with rollingmachining and a natural oxide layer).


28 Radiometric Properties of Materials

Transmission

Transmission is the process by which incident radiantflux leaves a surface or medium from a side other thanthe incident side (usually the opposite one). Spectraltransmittance τ(λ) is the ratio of the transmittedspectral flux to the incident spectral flux:

τ(λ)=Φλt/Φλi

Total transmittance τ is the transmitted spectral fluxweighted by the source function:

τ= Φt

Φi=

∫ ∞0 τ(λ)Φλidλ∫ ∞

0 Φλidλ

Note that total transmittance τ is not∫λτ(λ)dλ.

Transmittance may also be described in terms of radiance:

τ=∫ ∞

0∫ 2π

0 Ltλ

dΩtdλ∫ ∞0

∫ 2π0 Li

λdΩidλ

where Liλ

is the spectral radiance incident from direction(θi,φi), Ltλ is the spectral radiance transmitted to(θt,φt), and dΩ is the elemental projected solid angle,sinθcosθdθdφ.

Geometrically, transmittancecan be classified as specular,diffuse, or total, dependingon which direction is namedfor the emerging signal. Notethat the specular transmissiondirection has a spread to it: itdoes not occur exactly at one

angle. The bidirectional transmittance distributionfunction (BTDF) relates the transmitted and incidentbeams as

f t(θi,φi;θt,φt)= dL t(θt,φt)dE i(θi,φi)

= dL t(θt,φt)L i(θi,φi)dΩi

sr−1



Reflection

In reflection, a fraction of the radiation incident upon asurface is returned into the hemisphere above it. Exactlyhow much is returned, and its directional distribution,are determined by the wavelength and directionality ofthe incident radiation and the properties of the reflectingsurface.

Spectral reflectance ρ(λ) is defined analogously to itstransmittance counterpart, ρ(λ) =Φλr/Φλi, where Φλr andΦλi represent reflected and incident flux, respectively.Total reflectance ρ is therefore

ρ= Φr

Φi=

∫ ∞0 ρ(λ)Φλidλ∫ ∞

0 Φλidλand not

∫λρ(λ)dλ

Considerable research has been done on reflectanceprocesses due to their prevalence in many applicationsof optics and radiometry. The bidirectional reflectancedistribution function (BRDF) is the fundamentalgeometric reflectance descriptor:

fr(θi,φi;θr,φr)= dLr(θr,φr)dE i(θi,φi)

= dLr(θr,φr)L i(θi,φi)dΩi

sr−1

The equation describes the differential element of re-flected radiance dLr in a specified direction per unitdifferential element ofincident irradiance dE i

in a specified direc-tion. For a perfectly dif-fuse (Lambertian) sur-face, the BRDF is ρ/π.For a specular surface,BRDF is ρ/Ω, where Ωis the solid angle thesource subtends at thereflective surface.



Absorption and the Conservation of Energy

Absorption is the processby which a fraction ofthe incident radiant flux isconverted to another form

of energy, usually heat, and is neither reflected as flux,nor transmitted. Spectral absorptance α(λ) is defined asα(λ)=Φλa/Φλi, with the subscripts denoting absorbed andincident optical power, respectively. Total absorptance αis then

α= Φa

Φi=

∫ ∞0 α(λ)Φλidλ∫ ∞

0 Φλidλand not

∫λα(λ)dλ

Absorption removes power from a beam; directional char-acteristics such as bulk scattering are not often taken intoconsideration. Due to conservation of energy, the sumof transmitted, reflected, and absorbed beam componentsmust be unity:

τ+ρ+α= 1

The above statement assumes integration over all wave-lengths and directions. In the absence of wavelength-shifting effects (for example, Raman scattering or lumi-nescence), this relationship is also valid for any specificwavelength:

τ(λ)+ρ(λ)+α(λ)= 1

Reflectance factor R is the ratio of radiant fluxreflected by a particular sample to the radiant flux thatwould be reflected by a Lambertian surface under thesame irradiation conditions. Reflectance factor can beintegrated over a particular wavelength band or spectralquantity, R(λ). Although reflectance ρ cannot exceedunity, reflectance factor can take on values from 0 tonearly infinity—it is useful as a descriptor for diffusereflectors but not for other surface types.



Emission

The processes of transmission, reflection, and absorptionhave a key feature in common: all modify radiationthat is propagating. Emission is different in thisregard, as it is part of radiation generation rather thanpropagation. As a result, emittance is better defined interms of radiometric quantities we associate immediatelywith sources, radiance, or radiant exitance. Spectralemittance ε(λ) is the ratio of a source’s radiance to thatof a blackbody at the same wavelength and temperature:

ε(λ)= Lλ(source)/LλBB

Total emittance ε is also weighted by blackbody radianceand is a function of emission direction:

ε=Ðε(θ,φ;λ)LλBB sinθcosθdθdφdλ

(σ/π)T4

where σ= the Stefan–Boltzmann constant, and (σ/π)T4, tobe discussed later, is the integrated blackbody radiance atT. Note that total emittance is not

∫λ ε(λ)dλ.

One advantage of working with emittance is that it canbe equated to absorptance when conditions of thermalequilibrium are met. In that case, τ+ρ+α becomes τ+ρ+ε.In the infrared, most materials are opaque, and τ = 0,resulting in ε= 1−ρ. This relationship greatly simplifiesa variety of calculations, given appropriate geometries.

The wavelength-selective ratio α/ε is often employed incoating design, where it is desirable to have a surfacethat strongly absorbs in one spectral region and weaklyemits in another. This is the case in, for example, solarthermal systems used for water and space heating, inwhich coating absorption at low wavelengths is made ashigh as possible and emission at longer wavelengths aslow as possible.



Specular Transmissivity and Reflectivity

Specular transmissivity and reflectivity for a singlesurface may be calculated from the Fresnel equationsusing the complex index of refraction n + iκ. (Note thechange in terminology [“-ivity”] with reference to a “pure”substance rather than a specific sample.) In the case of noabsorption, κ= 0, the general equations are

ρp = (n2 cosθ1 −n1 cosθ2)(n2 cosθ1 +n1 cosθ2)2

ρs = (n1 cosθ1 −n2 cosθ2)(n1 cosθ1 +n2 cosθ2)2

τp = (2n1 cosθ1)(n2 cosθ1 +n1 cosθ2)2X

τs = (2n1 cosθ1)(n1 cosθ1 +n2 cosθ2)2X

where the ρi are reflectivity, τi are transmissivity, andX = (n2 cosθ2/n1 cosθ1). The two polarization states arerepresented by subscripts p (parallel) and s (perpendicu-lar, from the German word senkrecht). From these equa-tions, expressions in terms of θ1 may be derived usingSnell’s law and the law of specular reflection relating inci-dent and reflected beams, θr = θi.

ρp =

n1

√1−

(n1n2

sinθ1

)2 −n2 cosθ1

n1

√1−

(n1n2

sinθ1

)2 +n2 cosθ1

2

In the non-absorbing case, τp = 1−ρp and τs = 1−ρs.

ρs =

n1 cosθ1 −n2

√1−

(n1n2

sinθ1

)2

n1 cosθ1 +n2

√1−

(n1n2

sinθ1

)2

2



Single-Surface Illustrations

The total transmissivity τT or reflectivity ρT for un-polarized light is the average of the parallel and perpen-dicular components: ρT = (ρp + ρs)/2 and τT = (τp + τs)/2.Note that these are also called ρss and τss (single-surfacereflectivity and transmissivity, respectively).



More on Specular Propagation

Normal incidence greatly simplifies calculation with theFresnel equations, as θ1 = θ2 = 0. For a single surface andnon-absorbing medium, they are: ρss = [(n2−n1)/(n2+n1)]2

and τss = (4n1n2)/(n2 +n1)2.

Calculations become more complicated with an absorptivemedium. In these cases, the refractive index is the complexquantity

n+ iκ,κ(λ)= (λ/4π)α′(λ)

where α′(λ) is the material’s spectral absorption coeffi-cient (per unit length) and κ(λ) is the spectral extinc-tion coefficient. The internal transmittance τi of amaterial is described by an exponential absorption rela-tionship, the Lambert–Bouguer–Beer law

τi(λ)= e−α′(λ)x

where x is the path length. The product τo(λ)=α′(λ)x is thematerial’s optical thickness τo at that particular wave-length.

Optical thickness is dimensionless due to the product ofcm−1 and cm (or µm−1 and µm). Optical thickness discus-sions also apply to the atmosphere, where path lengthis in km. Spectral absorptance and spectral absorptioncoefficient are different quantities, the first being a mea-sured power ratio (an “-ance”), and the second a materialproperty needed to calculate an “-ivity.”



Transmission: Absorbing and Reflecting Materials

Real materials have bothabsorptive and reflectiveproperties; the transmittedradiation will be modifiedaccordingly. For a sampleilluminated by unpolarized,collimated radiation, thefirst-pass quantities are asfollows:

First-surface (single-surface) reflection: ρss =Φr1 /Φi

First-surface transmission: τss =Φt1 /Φi

Internal transmission: τi =Φ2/Φ1 = e−τo

For an infinite number of passes, the Fresnel equationsprovide the following relationships:

Transmissivity: τ∞ = (1−ρ2ss)τi

1−ρ2ssτ

2i

Reflectivity: ρ∞ = ρss +ρss(1−ρss)2τ2

i

1−ρ2ssτ

2i

Absorptivity: α∞ = (1−ρss)(1−τi)1−ρssτi

When the optical thickness is large, the material becomesopaque and the transmittance goes to zero. In this case,reflectance ρ approaches the single-surface reflectance ρss.When the optical thickness approaches zero, the materialbecomes transparent. In that case,

ρ= ρss +[ρss(1−ρss)2

](1−ρ2

ss); τ= (1−ρss)2

(1−ρ2ss)



Materials as Targets

Real targets (real materials) exhibit a variety of non-idealdirectional behavior; specular and Lambertian (perfectlydiffuse) are two extremes, but real targets do not reachthese limits and often exhibit a mix of both characteristicsin varying degrees.

Interestingly, when material transmittance is zero andthe index of refraction is complex, a polished metal(aluminum, above) exhibits Lambertian emissivity in thesum of both polarizations out to about 60 deg from normal,though the s- and p-polarization components vary fromthis. One would seldom consider a polished metal to beLambertian, and the amount of radiation emitted is low,but angular properties play a significant role.

On the opposite end in terms of emittance values, alu-minum, described as “weathered” or “anodized” in the lit-erature, is a diffuse emitter and has comparatively highinfrared emittance of 0.83–0.84.



Optical Material Selection Considerations

A system designer selecting optical materials mustconsider optical, thermal, mechanical, environmental, andother properties of the lenses, mirrors, housing, baffles,etc., used in the design. Many are listed below; not allapply in every case.

Material PropertiesOptical MechanicalTransmission Young’s modulusRefractive index Yield pointDispersion HardnessReflection Optical workabilitySub-surface scatter Coating compatibilityAbsorption Density, specific gravityHomogeneityBirefringence EnvironmentalFluorescence Solubility in water orAnisotropy other solvents

Surface deteriorationThermal Radiation susceptibilityThermal conductivity (space missions)Specific heat Behavior in vacuumHeat capacity Tolerance to vibrationCoefficient of linear thermal

expansion OthersSoftening point AvailabilityMelting point Safety

Cost

The phenomena that will be measured by the systemmust be considered, as well. Parameters include thespectral range of the phenomenon, its temperature,and its emittance. Some representative emittance valuesmeasured normal to the emitting object in the 8–14 µmregion of the infrared are supplied below.

Material Temperature EmittancePolished Al 300 K–900 K 0.03–0.04Heavily anodized Al 373 K 0.84Carbon, lampblack 300 K 0.95Human skin 300 K 0.98Water 273 K–373 K 0.96


38 Generation of Optical Radiation

Planck’s Law

Any object above 0K will radiate in some relationship toPlanck’s law:

Lλ = 2hc2

λ51

e(hc/λkT) −1

h = 6.626×10−34 J ·s(Planck’s constant)c = 2.998×108 m ·s−1

k = 1.381×10−23 J ·K−1

(Boltzmann’s constant)More commonly, Planck’s Law is stated as:

Lλ = c1

πn2λ51

ec2/nλT −1

c1 = 2πhc2 = 3.742×10−16 W ·m2

(first radiation constant)c2 = hc/k = 1.439×10−2 m ·K(second radiation constant)

The units of the above equation are W/m2sr m. Results areoften desired per nanometer or per micrometer, and caremust be used when converting from one unit to another.

Blackbody radiation occurs when the emittance ofthe radiating material over all wavelengths is 1, asin the above figure showing blackbody radiation forseveral temperatures. When the emittance is constant butnonunity, the material is said to radiate as a graybody.When the emittance is a function of wavelength, i.e., ε =ε(λ), the material is called a selective radiator. Althoughthere are no perfect blackbodies in nature, the model is agood one for many sources, including the sun.


Generation of Optical Radiation 39

Stefan–Boltzmann and Wien Displacement Laws

The Stefan–Boltzmann law provides the total radiantexitance from a blackbody source over all wavelengths:

M =σT4

σ = Stefan–Boltzmann constant, 5.67×10−8 W/m2 K4

T = temperature, in kelvin.

The radiant exitance of a greybody is its emittance ε,multiplied by this result. The radiance of a blackbody maybe obtained from total radiant exitance by dividing by π:

L = Mπ

The Wien displacement law (often referred to as Wien’slaw) provides the wavelength, in micrometers, of maxi-mum spectral radiance (or radiant exitance) of a blackbodygiven its temperature in kelvin:

λmaxT = 2898 µm ·KWhen blackbody spectral radiance (or radiant exitance) isplotted against wavelength for several temperatures on alog–log scale, Wien’s law is graphically represented by astraight line drawn through the curve peaks.



Rayleigh–Jeans Law and Wien Approximation

Two common approximations to the Planck equation areapplicable in certain circumstances. The Rayleigh–Jeanslaw, for long wavelengths and/or high temperatures, is:

Lλ = 2ckT

λ4

c = speed of lightk =Boltzmann’s constantT = temperature in kelvin

Note that Planck’s constant does not appear in the lawbecause it applies to wavelengths longer than thosefor which quantum effects must be considered. TheRayleigh–Jeans law is valid with less than 1% error ifλT > 0.778 m·K. The approximation is not used frequently,as less than 0.1% of blackbody output occurs at λT valuesgreater than 0.8 m ·K.

Of more utility is the Wien approximation, valid forshort wavelengths and/or low temperatures:

Lλ = c1

πλ5 e−c2λT

c1 and c2 are the first and second radiationconstants, respectively.

The Wien approximation is valid with less than 1% errorif λT < 3000 µm ·K, and is widely applicable.

The Rayleigh–Jeans law was dubbed the “ultravioletcatastrophe” by physicist Paul Ehrenfest due to itsprojection of infinite radiance at λ= 0.



Radiation Laws in Terms of Photons

As was the case for quantities expressed in terms of watts,all radiation laws have analogous expressions in terms ofphotons per second. In the case of vacuum (n = 1; thereforeλ= λ0):

Lqλ = c1q

πλ41

ec2/λT −1

Photon spectral radiance;c1q = 2πc = 1.883×109 m/s;Lqλ is in photons/ s m2 sr m

λq,maxT = 3700 µm·K Wien displacement law for photons

Lqλmax =σ′qπ

T4 Photon spectral radiance at thepeak;σ′q = 2.1×1011 s−1 m−2 K−4 µm−1

Lq = σq

πT3

Total photon radiance(Stefan–Boltzmann law);σq = 1.52×1015 s−1 m−2 K−3

The locus through the peaks of the curves representsWien’s displacement law.

In some cases, it is desirable to use photon spectralradiance rather than its power-related sister distribution.For example, problems for which photon detection isrequired make good candidates for photon radiancedistribution usage.



Kirchoff’s Law

A small blackbody en-closed in an isother-mal (∆T = 0) cavity willhave the same temper-ature as its enclosurewhen the system isat thermal equilibrium.Absent additional radi-ation sources or sinks,the power emitted by

the source must equal the power it absorbs: αΦi = εΦBB.Because the incident and emitted power are the same,

α= ε,which is Kirchoff’s law. However, in addition to thermalequilibrium, spectral and directional constraints mustalso be applied to the above expression.

Directional Spectral α(λ;θ,φ,T)= ε(λ;θ,φ,T)No constraints other than thermal equilibrium

Directional Total α(θ,φ,T)= ε(θ,φ,T)(1) Spectral distribution of incident power proportionalto BB at T, or(2) α and ε independent of wavelength

Hemispherical Spectral α(λ,T)= ε(λ,T)(1) Incident power independent of angle, or(2) α and ε independent of angle

Hemispherical Total α(T)= ε(T)(1) Incident power independent of angle and spectraldistribution proportional to BB at T; or(2) Incident power independent of angle and α, εindependent of wavelength; or(3) Incident power at each angle has spectraldistribution proportional to BB at T and α, εindependent of angle; or(4) α and ε independent of both angle and wavelength(Table derived from Ref. 2)



Natural Sources

Sources of radiation may be classified in a variety of ways,including natural or artificial. Among natural sourcesare the sun, moon, the earth, zodiacal radiation, and evenhumans. Each of these sources may be further classified asactive or passive, depending on whether they generateradiation, as the sun does, or reflect it, like the moon. Somesources are both active and passive: zodiacal light resultsfrom solar reflection off dust in the visible spectrum; inthe infrared, the radiation mechanism is thermal emissionfrom dust particles.

E r

−2 nm−W mGlobal il Direc +circumsolar

−2 nm−W m−2 nm−W m

The sun is our primary source of natural radiationon earth. For applications such as solar photovoltaics,direct solar illumination and global illumination (from ahemisphere) are often considered.

The above plot utilizes a portion of the ASTM G-173-03reference spectrum for an airmass of 1.5. Airmass isdefined as the secant of the solar zenith angle, with anairmass of 1 indicating that the sun is directly overhead.Airmass 1.5 implies a solar zenith angle of about 48 deg oran elevation angle of about 42 deg.

The earth (300 K) becomes a primary source of radiationin the IR, particularly in the atmospheric window8–12 µm.



Lambert–Bouguer–Beer Law and Langley Plot

The Lambert–Bouguer–Beer law describes attenuationnot only within optical media but also within theatmosphere as well. In this application it expressesthe wavelength-dependent irradiance received at a pointon earth as a function of the exoatmospheric spectralirradiance:

Eλ = Eoλe−τo(λ)m

where Eoλ = exoatmospheric spectral solar (or stellar)irradiance, m = airmass, and τo(λ) = atmosphere’s opticalthickness at that wavelength.

The law is strictly valid only monochromatically and as-sumes an unchanging atmosphere during the measure-ment period. Note that Eλ/Eoλ is also called transmit-tance.

This information can be transferred to a Langleyplot, which expresses irradiance at one wavelength asa function of airmass. The slope of the line is thespectral optical thickness τo(λ); the y intercept is theexoatmospheric solar irradiance at that wavelength thatmay be obtained by extrapolation. τo(λ) is also calledoptical depth:

τo(λ)= e−α′(λ)x

The absorption coefficient may be further divided intomolecular and aerosol scattering, and absorption compo-nents.



Artificial Sources

Natural and artificial sources may be classified aseither thermal or luminescent. Thermal sources emitradiation as a function of temperature or reflect it;luminescent sources emit radiation via atomic transitions.Some sources are both thermal and luminescent. Artificialsources vary widely in type and spectral characteristics;measurements must always take into account spectral,directional (geometric), and spatial properties of theradiation.

Thermal sources Luminescent sourcesIncandescent lamps LEDs, IREDs, OLEDsHot metals PhosphorsExhaust gases Exhaust gases

(CO2, CO, H2O) (CO2, CO, H2O)Electrical wiring Fluorescent lamps, CFLsGlobar, carbon arc LasersSoot Laser diodesOptical elements Sodium vapor lampPaint Mercury vapor lampGlass Xenon arc

Exhaust gases, such as from diesel engines or muzzleflash, emit radiation at wavelengths where they are opti-cally thick and often include the blackbody contributionof particulates >30 µm in diameter.

If a gas spectrally absorbs where it emits, how can wesee it? Due to higher temperature and pressure, a hotgas will have a slightly different spectral profile than anunheated one. While the center wavelength will absorb,the hot gas may be observed in the two wavelength re-gions bordering the center, as a function of temperature,pressure, spectral region, and emitting species.



Luminescent Mechanisms

Although temperatures mustbe high to achieve signif-icant radiation from ther-mal sources in the visibleand UV (the sun, for exam-ple, is often modeled as a6000 K blackbody), lumines-cence generates radiation atlower temperatures. Doppler

(Gaussian) and pressure (Lorentzian) effects providebroadening of spectral lines, so that emission is not strictly“monochromatic,” but it exists in a narrow spectral re-gion. Luminescent sources tend to be more efficient thanthermal sources in generating light, as their spectra canbe engineered to fall in the visible band with little or noinfrared.

Doppler Effects Lorentzian EffectsLow-pressure discharge Pressure-broadened linesLittle power in each line High radiant power, efficientSharp line applications Wings combine to formWavelength calibration continuum–Fluorescence excitation –Illumination, searchlights–Interferometry –Projection systems

–Solar simulatorsLow-pressure lamps High-pressure lamps–Hg, He, Ne, Na, K, –Hg, Xe, Hg-Xe, Na

Zn,Cd, Cs –Can be modulated andflashed

A low-to-moderate-pressure mercury discharge source(254-nm line) becomes a fluorescent source with theaddition of a phosphor, allowing the UV photons absorbedby the phosphor to cause emission of visible photonswe see as light. Compact fluorescent lamps (CFLs)are higher-efficiency variants packed into a smallervolume. They are increasingly used in many householdapplications in place of incandescent (tungsten-filament)lightbulbs.



Some Luminescent Sources

Light-emitting diodes (LEDs)utilize a forward-biased p-n junc-tion to generate radiation. Theemission comes from recombina-tion radiation at the band edgeexcited by high current density.The emission spectral region isnarrow, about 5% of the centerwavelength. LEDs are small, con-sume little power, and have life-times greater than 10,000 hrs.They are finding increasing usein lighting applications, replac-ing incandescent sources in traf-fic and roadway lighting, automo-biles, and airline cockpits. Theyare available in a variety of wave-lengths.

Red, green, and blue LEDs maybe combined to produce a “white”LED. White LEDs may also beproduced by using a blue LED in conjunction with aspecific phosphor (such as Cs:YAG) to produce radiationover a broader region from about 500 to 700 nm.

Organic light-emitting diodes (OLEDs), a class ofLED made from polymers and other organic materials,have very thin width and are competing with LCDs as“sheet displays” in many applications.

Lasers (an acronym for Light Amplification by StimulatedEmission of Radiation) operate due to photon stimulationor induction of an exicted-state molecule to induce asecondary photon emission, in phase with the initialphoton. As a result, lasers are coherent sources, and thepower transfer laws detailed herein generally do not applyunless the laser is coupled to an integrating sphere toproduce a diffuse beam.


48 Detectors of Optical Radiation

Detector Types

Detectors of optical radiation may be classified severaldifferent ways: there are point and area detectors,thermal and photon detectors, and various kinds ofarrays. The responsivity R (A/W or V/W) of a photon orthermal detector is the ratio of output electrical signal toincoming radiant power:

R=∫

R(λ)Φλdλ∫Φλdλ

Both power and responsivity maybe expressed as spectral or totalquantities. Ideal responsivities areshown below.

Photon responsivity Rq is defined analogously, withthe input being photon flux (photons/s) and the output acurrent or voltage.

The cutoff wavelength λc is the wavelength at which adetector ceases to be responsive.

There are several significant differences between photonand thermal detector categories in addition to the shape oftheir response curves.

Parameter Thermal PhotonSpectral response Wide NarrowDetectivity Low HighResponse time Slow FastCooling Not typical Typical in IRChopping Often RarelyCost (IR) Low High


Detectors of Optical Radiation 49

Detector Definitions

Cutoff frequency fc is the electrical frequency, in Hz, atwhich the detector response falls to 0.707 of its midbandmaximum value. It is also called 3-dB frequency.

Time constant τ is the time required for a signal toachieve 63% of its final output, given a step input:

τ= 12π fc

Thermal time constant τT is defined analogouslyto electrical time constant, 1/(2π fT ). It characterizes athermal detector’s speed of response, with fT called thethermal cutoff frequency.

rms (root-mean-square) value (voltage or current) isdefined mathematically as

vrms =√

1T

∫ T

0v2(t)dt

where T is a single or integermultiple period, for a periodicwaveform.

Responsive quantum efficiency η is the number ofindependent output events per incident photon, oftenexpressed in electrons/photon. It is also called quantumefficiency.

Curie temperature Tc is the temperature at which aferroelectric material loses its polarization. It is importantin pyroelectric detectors.

It is always important to note the units of responsivityR, particularly since the same term is used to describethe radiometric performance of an entire system. Inthe latter cases, irradiance responsivity RE andradiance responsivity RL are commonly seen.



More Detector Definitions

Signal vs or is is the rms component of detector outputvoltage or current arising from a specific radiometricinput.

Noise vn or in is the rms component of detector outputvoltage or current arising from random fluctuations in thedetector circuit, incoming radiation, and other factors.

SNR= vs

vn= is

in

Signal-to-noise (SNR) ratio (in am-plitude) may be defined in terms ofeither voltage or current. It is dimen-sionless.

Z = dVdI

Impedance Z (ohms) is the slope of thecurrent versus voltage curve of a device at thedesignated operating point.

RA product is the product of a detector resistance and itsarea and is a constant for many materials.

BLIP is an acronym for Background-Limited InfraredPhotodetector, one for which the limiting noise in thedetector output arises from background (not signal)photons.

Effective noise bandwidth B (also ENB and ∆ f ) isan equivalent square-band bandwidth, differing from the3-dB bandwidth.

B = 1G( fo)

∫ ∞

0G( f )d f

For spectrally flat (white) noise, G = power gain f0 = fre-quency where G is maximum.

Advantage of ENB:total noise powercan be obtained bymultiplying ENB bythe power spectraldensity, as long asthe latter is con-stant.



Detector Figures of Merit

Detective quantumefficiency DQE is the ratioof the squares of output andinput SNR.

DQE= SNRout2

SNRin2

Noise equivalentpower NEP (W) is the inputpower producing a unitySNR.

NEP= vn

R=Φvn

vs

Detectivity D (W−1) is thereciprocal of the NEP.

D = 1NEP

= SNRΦinput

Specific detectivity D∗(cmHz1/2W−1) is Dnormalized for detector areaand noise bandwidth B.

D∗ = D√

AdB =√

AdBNEP

=√

AdB

Φ

vs

vn

D∗∗ (cmHz1/2W−1) normalizesD∗ to a π sr projected solidangle FOV. It is pronounced,“Dee double-star.”

D∗∗ = D∗√Ω

π= D∗ sinθ1/2

All figures of merit have both spectral and photon for-mulations, e.g., D∗(λ) is D∗ at a specific wavelength:

D∗(λ)=√

AdB

NEP(λ)cm ·Hz1/2/W

Photon D∗ is the specific or normalized detectivity interms of photons:

D∗q = D∗ hc

λ=

√AdBn

is

incm ·Hz1/2/(photons/s)

where n in the denominator is the photon flux in photonsper second at wavelength λ.



Noise Concepts and Definitions

Noise can be expressed as a voltage, current, or power,and is best expressed as a spectral (per Hz) quantity ifits magnitude depends on frequency. The mean-squarenoise voltage (equivalent to noise power) is:

v2 = (vi −vavg)2 = 1T

∫ T

0(vi −vavg)2dt

where vi = instantaneous voltage, vavg = average voltage,and T = observation time.

i2 = v2

R2

Current may be used with thesubstitution (Ohm’s law):

Independent (uncorrelated) noise powers add alge-braically; noise voltages or currents add in quadrature.

v2n = v2

1 +v22 +·· ·+v2

i

vn =√

v21 + v2

2 +·· ·+v2i

Noise factor F is the ratio of a detector’s actual noise tothe theoretical limit.

F = real noise powerideal noise power

= SNRin

SNRout= 1√

DQE

Noise figure NF is the log10 of the noise factor. Expressedin dB, it is

NF = 10× log[

real noise powerideal noise power

]dB

Noise temperature Tn is the temperature of a thermalsource providing a signal power equal to the noise power.

Tn = v2n + i2

nR2

4kRB



The Most Unpleasant Noises

Johnson noise (Nyquist,thermal) occurs due torandom motion of carriersin any electrical conductor.The second expression isvalid when B is establishedby an RC circuit timeconstant.

v j =p

4kTRB; v j =p

kT/Ck = Boltzmann’s constant1.38×10−23 J/K

T = Temp (K)R = resistance, ΩC = capacitance, F

Shot noise occurs due tothe discrete nature ofelectronic charge (as invacuum tubes, for example)seen when current flowsacross a potential barrier.

is =√

2qIdcBq = elec. charge,1.6×10−19 C

Idc = direct current acrossa potential barrier

Generation–Recombina-tion (G–R) noise arisesfrom fluctuations in therate at which chargecarriers are generated andrecombined withinsemiconductors. Theequation form depends ondetector configuration.G–R is not present inthermal detectors.

vG–R = 2iR

√τlB

N[1+ (2π f τl)2](extrinsic photoconductor)N = mean number ofcarriers

i = average photocurrentdue to Nτl = carrier lifetime

1/f noise is insidious noisefrom various sources. It isubiquitous and does notdecrease throughintegrating over time.

i1/ f =√

KsIαdcB

fβ

1.25<α< 4 (typically 2)0.8<β< 3 (typically 1)Ks is source dependent

Temperature fluctuationnoise occurs due tomicrofluctuations inthermal detectortemperature. It is relatedto the statistical noise inblackbody radiators.

∆T =√

4kT2KB

K2 +4π2 f 2H2

K = thermal conductance(W/deg)

H = heat capacity (J/deg)



More Unpleasant Noises

Photon noise arises because photons randomly arriveat the detector. Photons obey Poisson statistics, so the√

(∆n)2 =√

nmean-square fluctuations in the photonarrival rate are equal to the arrival rate,with n the number of photons.

Microphonic noise is acoustically generated. It isfound in pyroelectric thermal detectors, which act likemicrophones. Unanchored wiring in a vacuum dewar alsoproduces this noise.

Triboelectric noise originates from charges built upwithin dielectrics, typically within coaxial cables.

Amplifier noise is a form of 1/ f noise typically foundin low-level circuitry coupled to the detector output. It isusually related to f 2.

Quantization noise occurs when analog signals are dig-itized. It is related to the number of bits n in the digital

LSB= SIGNALmax

2n

word, with LSB the magnitudeof the least significant bit andSIGNALmax the full-scale signal.

CCD noises arise from processes related to chargetransfer and readout in CCD arrays, including variationsin charge transfer efficiency (CTE), readout noise, andkT/C noise arising from the residual charge left on anintegrating capacitor after a reset switch is closed andopened.

For a photoconductive detector, 1/ f , G–R, and Johnsonnoise voltages are uncor-related (independent) andadd in quadrature.



Thermal Detectors

In thermal detection, incident radiation is absorbed bythe detector’s receiving surface, causing an increase insurface temperature. The temperature change is detectedby one of several means, giving rise to thermoelectricdetectors, resistive bolometers, pyroelectric detectors,and others. All thermal detectors operate according tothe same relationship between incident radiation, thedetector, and its immediate surroundings:

∆T = Td −To = αΦRT√1+ω2R2

T H2

Td =detector tempTo =heat sink temp

RT = thermal resistance between receiver and heat sink(deg/W)H = heat capacity of receiver (J/deg)ω = radian frequency of incoming signalα = detector material absorptanceΦ = incident radiant power Φ0e jωt to allow for eithersteady-state or modulated (chopped) radiation.

Thermal time constant τT = RT H

Thermal resistance RT = 14ασAdT3

dσ = Stefan–Boltzmann constant

Thermal conductance K = 1RT

In the theoretical limit, noise occurs due to fluctuation ininput signal power (analogous to photon noise, for photon

NEP= 4

√kT5

dσAdB

α

detectors). In practice, this limit israrely achieved, and the detectoris typically Johnson-noise limited.



Thermoelectric Detectors

Thermoelectric detectors operate according to the ther-moelectric effect, in which a temperature differenceproduces a voltage dif-ference (and vice versa).Given two dissimilarmetals connected in se-ries (a thermocouple)and placed in a circuit,a current will flow inthe direction indicated

S = ∆V∆T

(V/deg)

if temperature T2 in metal 2 is greaterthan T1 in metal 1. Opening the circuitto check the voltage due to current flowyields S, the Seebeck coefficient orthermoelectric power.

Π= 1I

(dQdt

) The Peltier effect occurs when currentis passed through a thermoelectric circuit,resulting in one of the junctions between

the two materials becoming warm while the othercools. The Peltier coefficient Π describes the effect’smagnitude in terms of heat flow (dQ/dt) and current I.This effect is exploited in thermoelectric coolers used tocool detectors, laser diodes, dew-point sensors, and otherdevices.

∆T =αΦRT

(1− RT S2Td

R

)The Peltier effect also causesa reduction in sensitivity (re-duced ∆T). The signal, how-ever, may be increased by placing several junction pairs inseries to form a thermopile. Johnson noise predominates.

Material S (µV/oC) Material S (µV/oC)Al −0.5 Bi −60Cu +2.7 Sb +40Ag +2.9 Fe +16



The Bolometer

A thermoresistive materialthat absorbs radiation be-comes warmer, and the re-sulting change in its electri-cal resistance can be sensedusing a bolometer, a resis-tor possessing a high temper-ature coefficient of resis-tance (TCR). TCR is denotedby β and is in units of K−1.Bolometers may be fabricated

from metals (classical) or semiconductors (more recent).

To measure resistance, a current must be forced throughthe circuit; the half-bridge configuration is common forthis device, with the coupling capacitor used for blockingthe DC signal across the sensitive component RB.

Signal voltage measured across the terminals is

Vs =∆V = VBRL∆RB

(RB +RL)2

If RL = RB (maximum power transfer theorem),

VS =∆V = VBR∆R

(R+R)2=

(VB

4

)(∆RR

)=

(VB

4

)(Roβ∆T

R

)The bolometer voltage responsivity is then:

Rv = Vs

Φ=

(VB

4

) αβRT√1+ω2τ2

T

Relatively low-cost infrared cameras using uncooledsemiconductor microbolometer arrays have becomeincreasingly popular in recent years. It is not uncommonin today’s (2011) market to see cameras with less than20-µm pixel size in a 640×480 array.



Pyroelectric Detectors

The pyroelectric effect displayed in some ferroelectricmaterials is a change in surface charge with temperature.This property is used in the pyroelectric detector, whichis capable of high-speed operation and responds only tochanges in the input signal (which requires chopping).The pyroelectric coefficient p is the change in electricpolarization due to a change in temperature:

p = dPs

dTC/cm2K

p increases until the material’s Curietemperature is reached, above which itdrops to zero.

Rv = αωpAdRLRT√1+ω2τ2

√1+ω2τ2

T

The voltage responsivity ofpyroelectric detectors is afunction of both thermal and

electrical frequencies, the pyrolectric coefficient, thermalresistance, and load. There is the typical speed–sensitivitytradeoff issue.

Material p (C/cm2 K) D∗ (cmHz1/2/W) RemarksTGS 4×10−8 109 High D∗

LiTaO3 2×10−8 6×108 “Bulletproof” 1

SrBaNbO3 6×10−8 5×108 FastestPVF2 2×10−8 2×108 Cheap

1 James M. Palmer (1937–2007)



Photon Detectors

Photon detection re-lies on materials thatallow an incident pho-ton with sufficient en-ergy to elevate an elec-tron from a bound state(valence band) to afree state (conductionband) where, depend-

ing on the detection mechanism, it is available for conduc-tion under an applied electric field or (in the photoemissivecase) for escape into vacuum.

The energy of the incident photon must be greater than orequal to the gap energy Eg, and the cutoff wavelength(µm) for this process may be expressed as λc = 1.24/Eg

with energy in eV. The difference between Ep and Eg

constitutes thermalization loss and causes the voltageand power in a solar photovoltaic cell to decrease. E f isthe Fermi level, which has a 50% probability of beingoccupied by an electron.

Materials useful in photon detection include intrinsicsemiconductors having only a small concentration of im-purities and extrinsic semiconductors into which im-purities have been introduced via the doping process toextend the wavelength response.

Material Energy gap (eV) λc (µm)SiC 3.0 0.41CdS 2.4 0.52GaP 2.25 0.55Si 1.12 1.1Ge 0.68 1.8PbSe 0.26 4.8Hg(1−x)CdxTe variable 1 to 24



Photoconductive Detectors

The conductivity σe of aslab of semiconductor ma-terial is a function of elec-tron concentration, hole con-centration (the positions leftin the valence band afteran electron moves into theconduction band), electroniccharge q, and mobility of

electrons and holes. When the detector based on this ma-terial is placed into a circuit and bias voltage applied, avoltage responsivity can be computed:

Rv = IdcRληqµτl

2hczAdσe

where η = quantum efficiency, µ = carrier mobility (gener-ally the sum of electron and hole mobilities), τl = carrierlifetime, and σe = electrical conductivity (Ω ·cm)−1.

The photoconductive gain G of the device is the ratio ofthe carrier lifetime to the carrier transit time τtr betweenthe electrodes.

G = τl

τtr

Noise current in photoconductive detectors is:

in2 = 4q

[ηqΦ

(λ

hc

)G2 + qG2N ′+ kT

qRD+ kT

qRL

]B

Terms inside the bracket, according to order:

1. G–R noise from incident photons (signal and back-ground)

2. Dark current noise due to N ′ thermally generatedcarriers

3. Johnson noise in the detector resistor, RD and4. Johnson noise in the load resistor RL

D∗BLIP (λ, f ) = λ/2hc × (η/Eq)1/2, where Eq is photon inci-dence.



Photoemissive Detectors

The process of photoe-mission relies on an ex-ternal photoeffect, inwhich an electron re-ceives sufficient energyfrom an incident photonto physically escape aphotosensitive materialcalled a photocathode.

In this three-step process, a photon is absorbed, resultingin a “hot” electron; the electron moves to the vacuum in-terface; and the electron escapes over the surface barrierinto the vacuum, where, in practical devices, it is capturedby a positively charged anode.

The work function φ is the energy required for an elec-tron to escape the surface barrier. For both metals andsemiconductors, it is used to calculate the cutoff wave-length according to λc = 1.24/φ with λ in micrometers andφ in eV.

An expression for noise voltage is:

vn = RL

[(2qid +2q2ηΦ

λ

hc+ 4kT

RL

)B

]1/2

Terms inside the bracket:

1. Shot noise due to dark current id2. Shot noise due to signal + background current3. Johnson noise in load resistor RL

Johnson noise and dark current are reduced by cooling,leaving signal-dependent shot noise as the main noisesource.

Current responsivity for a photocathode: Ri = qηλ/hc

Prominent photoemissive detectors are the photomulti-plier tube (PMT) and microchannel plates.



Photovoltaic Detectors

The photovoltaic detector responds to incoming pho-tons striking a p-n junction in a semiconductor mate-rial. The junction is formed by doping adjacent areas dif-ferently so that one becomes an n-type (donor) and theother a p-type (acceptor) of electrons. The physical areabetween these two sections is called the depletion re-gion, which contains an electric field due to ionized donorsand acceptors.That electricfield creates apotential dif-ference φ0 be-tween p and nregions, shift-ing the bandenergies of thetwo materials.

If a forward bias is applied to the p region, the potentialbarrier height is reduced, current flow increases, and thedepletion region narrows. If a reverse bias is applied tothe n region, the barrier height is increased, the currentdecreases, and the depletion region widens.

Photovoltaic devices are often used as power generators,as in the case of a solar cell.



Photovoltaic Current and Performance

In the absence ofoptical radiation onthe photodiode, thereis no photocurrentthrough the detector,and the detector’sdark level is definedby the reverse sat-uration current I0.Incident optical ra-diation generates acurrent through the diode, shifting the current-voltage(I-V) curve downward. The expression for total currentthrough the diode becomes

I = Io

(e

qVβkT −1

)− Ig

with Ig = photocurrent, β = “ideality factor” that varieswith applied voltage—1 for the ideal case, but 2–3 for thereal case. The currentresponsivity of photo-voltaic detectors is Ri =ηqλ/hc.

The detector is shot-noise dominated. Noisecurrent may be expres-sed as a function of thesignal current in the visible spectrum: ishot = (2qisB)1/2. Inthe infrared, shot noise current is a function of the back-ground:

ishot =[2ηq2 λ/hc ΦB

]1/2

Johnson noise in the load resistor and 1/ f noise are alsopresent but play less significant roles.

D∗BLIP(λ, f )= ηλ/(2Ebackhc)1/2

where Eback is the background irradiance.



Detector Interfacing

A detector of optical radiation functions as a transducer,converting one form of energy to another to produce anoutput electrical signal. Amplifiers operate on the outputsignal to bring it up to a level compatible with the circuitryused in the electrical subsystem.

Calculations of the SNR out of an amplifier must take intoaccount the value of the feedback resistor (if applicable)and various noise sources, which may include Johnsonnoise, shot noise, and 1/ f noise, along with noise in theamplifier itself.

SNR = signal voltage out / noise voltage out. In thiscase, where the pre-amp may be coupled to the outputof a photovoltaic device, Eout = −IpcR f , where Ipc is thephotocurrent from the detector and R f is the value of thefeedback resistor.

More than one detector type may utilize a particularamplifer type, as seen in the case of the PMT.



Single and Multiple Detectors

The detection principles discussedthus far have been specifically ap-plied to single-element detectors;they are applicable to multipleelement detectors as well. Arraydetectors may be linear or (morecommon in most imaging today)two-dimensional. They may beutilized with a scanner or in staring mode. A mechanicalscanner provides an additional source of system noise, butthe use of multiple detectors increases SNR by the squareroot of the number of detectors, SNRnew = SNRold × N1/2.Point sources underfill the FOV of an individual detectorelement (pixel), whereas extended sources overfill it.

This instantaneous field of view(IFOV) is often measured in micro-radians for many of today’s high-performance imagers, and it deter-mines a system’s spatial resolution.For instance, if the IFOV is 3 µradand the system is at 300 km altitude, the spatial resolu-tion is 0.9 m on the ground.

Single Detector Multiple Detectors

Information gathered fromone λ band or FOV at onetime

Simultaneousmeasurements possible

Simple electronic interface More complex electronicsdue to array format

Complex scanning schemeneeded due to one detector

Simple scan or no scanrequirement if a staringarray

One detector, one response Magnitude of response canvary over the array dueto responsivity nonuni-formities between pixels



Detector Array Architectures

Further tradeoffs may be made among array technologies,including CCD, CMOS, and CID. Charge-coupleddevice (CCD) technology achieves arrays with largenumbers of pixels and allows efficient charge transferto the array output. CMOS (complementary metal-oxide semiconductor) structures cost less than CCDs tomanufacture and exhibit high yields. Charge-injectiondevices (CIDs) employ pixels that may be addressedindividually. They may be used to stare at scenes for longperiods without charge spillover (“blooming”) into adjacentpixels.

CCD CID CMOS

Array fillfactor

100%possible

80% orbetter

Varies:30%–100%

Sensitivity Highest High whenintegratingframes

Not as highas CCD

Pixel-to-pixeluniformity

High Not as highas CCD

Can be highfor on-chipprocessing

Bloomingcontrol

Can be aproblem

Excellent Can beimproved

Fabrication Requiresspecializedprocessing

UsesstandardCMOSprocessing

Usesexistingsemicon-ductor fabtechniques

Advantages Well-establishedandcharacterized

Low-lightlevel apps& excellentbloomingcontrol

Manyon-chipsignalprocessingfeatures

Disadvantages Bloomingcontrol

Noisy Readoutscan benoisy



Choosing a Detector

Several practical questions arise when choosing a detectorfor a particular application, among them:

• Is the application imaging or non-imaging?• What is the desired spectral resolution?• What is the desired spatial resolution?• What dynamic range is required?• What is the expected temporal response of the phe-

nomenon you wish to detect or measure?• How do expected system size, weight, and power fac-

tor into the choice of detector?• What is the cost of the detector (or array)?

There are other questions, but these are the first thatmight be asked. In all cases, the first considerationshould be the physical characteristics of the phenomenonunder study/measurement, with practical considerationsincluding cost constraints to follow.

Designers can err by choosing a detector that does notspectrally match the phenomenon to be measured.

In the first case, the detector’s filtered response does notmatch the wavelength range of the source; in the second,the wavelength response is matched by the filtered detec-tor to within a small error. However, if the goal of theproject is to collect information from the source in only thesmall spectral region indicated in the first figure, then thefirst filtered detector would be a good choice.


68 Radiometric System Components

Choppers and Radiation References

Many choppers are availablethat modulate the input signalwith respect to time for detec-tors, including pyroelectrics. Themost common are spinning bladesdriven by an electric motor. Chop-pers allow the system designer touse drift-free ac amplifers and toavoid the 1/ f noise region. Disad-

vantages include loss of at least 50% of the input power,the addition of noise both electronic and acoustic, andthe possibility of unintended reflection from blades ab-sent proper coating for the wavelength region in which thechopper operates.

The fraction of radiant power available to the detectoror system after chopping may be calculated as thechopping factor Fc, which is waveform dependent—fora square wave input, it is about 0.45, and for a sine waveinput (most common), it is about 0.35.

Radiation reference sources may be contained withinan instrument or in the instrument’s environment (testsetup or flight platform). They are of two kinds: (1) in-ternal calibration sources and (2) references whose knownproperties allow the correct target signal to be deduced.In (2), Correction Factor = Signaltarget −Signalref , whereSignaltarget is the uncorrected target signal and Signalrefis the internal reference signal. The correction factor isthen algebraically added to the target signal to producethe corrected output. Ideally, the reference signal shouldbe from a source very similar in radiative character to thedesired target. In the infrared, referencesinclude a blackbody or a blackened or re-flective chopper. Solar diffusing panelsmay be used as radiation references andas on-board calibrators in the solar-reflective wavelengthregion (VIS–NIR particularly).


Radiometric System Components 69

Baffles and Cosine Correctors

Baffles are mechanical structures within an instru-ment that reject out-of-field radiation from stray-lightsources. In the best cir-cumstances, baffles arearranged so that the de-tector does not “see” di-rectly illuminated sur-faces.

Simplified Procedure for Baffle Placement

1. Draw segments A-B (locus of aperture edges).2. Draw A-A′ (edge of detector to front corner).3. Draw Baffle #1 (intersection of A-B and A-A′).4. Draw B-B′ (edge of aperture through Baffle #1).5. Draw A-B′ (edge of detector to front aperture shadow).6. Draw Baffle #2 (intersection of A-B and A-B′).7. Repeat.

Baffle materials should be specular, as the goal is to defineexactly where the stray light is heading.

Cosine correctors accept radiation from a wide angularrange, enhancing the spatial capabilities of a receiver,fiber optic probe, or detector head. The desired resultfor this transmitting diffuser is to approach E = E0 cosθ

is closely as possible, where θ is the angle betweenthe surface normal and incoming radiation, and E0 isirradiance at normal incidence.



Spectral Separation Mechanisms

Spectral instruments employ a dispersive elementto separate component wavelengths of incoming opticalradiation. Two of the most common are spectrometersand spectroradiometers, the former providing awavelength-dependent distribution of incoming signaland the latter anchoring that distribution to a specificradiometric quantity (such as irradiance).

The heart of the instrument is the monochromator withits dispersing element. Prisms and gratings are the mostcommon dispersers, but others are used as well.

Spectroradiometers are characterized in terms of resolv-ing power Qrp and throughput. The resolving powerrelates a physical parameter of the disperser to λ/dλ,where λ is the center wavelength of a band, and dλ is theminimum wavelength separation for two line sources cal-culated according to the Rayleigh criterion, 1.22λ( f /#).Expressions for throughput include the area of theentrance slit and the projected area of the disperser.

Do not overfill the spectrometer slit—there is no advan-tage in throughput to be gained, and the disadvantage ofpossible stray-light effects exists.

Spectroradiometers may employ either single detectorsor arrays at the focal plane; there are advantages anddisadvantages to each approach.


Radiometric System Components 71

Prisms and Gratings

For a prism-based instrument,Qrp = λ/dλ = bdn/dλ, where bis the length of the prism baseand dn/dλ is the change in theprism’s refractive index as a

function of wavelength. Throughput T is

AΩ= (Aslit Aprism cosθ)/ f 2

where the area of the slit is normal to the optical axis, thecosθ factor accounts for the projected area of the prism,and f is the focal length of the focusing optical element.

A grating spectrometer obeys the grating equation, mλ=d(sinθi + sinθd), where λ is the wavelength of optical ra-diation, θi is the angle of incidence, θd is the angle ofdiffraction, and d is thespacing between grooves.The integer m is the or-der number, which ac-counts for the periodicityof the diffracted wave-form resulting in multi-ple diffraction patternsat each wavelength. Resolving power of a grating instru-ment is Qrp = mN, where N is the number of grooves illu-minated. The throughput expression is similar to that of aprism instrument: T = (Aslit Agrating cosθ)/ f 2. In this case, θ

also refers to the angle between the optical axis and grat-ing normal.

Prism GratingThroughput Low HigherEfficiency ∼ Constant Strongly peakedMonochromaticity Single λ Multiple ordersDispersion (dθd/dλ) Nonlinear LinearBandwidth (fixedslit)

Narrow inblue, wide inred

Constant



Filters

Filters may be used to providespectral selection in a radiometricinstrument. They do so by passingsome wavelengths while rejectingothers. Thin film interferencefilters are common for this purpose.They function as optical resonantcavities that permit only certainwavelengths to be transmitted when irradiated by abroadband source. When deployed off normal, theirpassband shifts and broadens, providing a reduced peakat a shorter wavelength.

Absorption filters con-structed of gels or glassalso provide spectral se-lection, although theytypically have largerbandwidths than thenarrowest interferencefilters. As the angle ofincidence of incomingradiation is increased, they are not as prone as inter-ference filters to change in the cut-on and cut-off wave-lengths. Both absorption and interference filters can beused in a filter wheel to provide a (discontinuous) sourcespectrum, with each filtered channel providing one spec-tral value.

Neutral density (ND) filters attenuate an incomingsignal. They find many uses in photography to reducethe illuminance of a scene and are also used in scientificapplications. The relationship between filter number andattenuation is

n =− log(Efiltered/Eincident)

For example, filter ND 2 reduces signal irradiance by afactor of 100.


Calibration and Measurement 73

Radiometric Calibration Basics

Radiometric systems are calibrated to tie the system’soutput signal (a digital count value, voltage, or current)to an input quantity such as radiance, irradiance or(less often) optical power. Calibrations may be absolute—tied through a chain of traceability to a NIST standard, for

example—or relative,relying on correlationbetween detector re-sponses in an array,dataset characteristics,features in a scene, orother parameters.

In the absolute case,the calibration curveassociates sensor out-put with a specific ra-diometric input.

It is always desirable to make the calibration equationas simple as possible—a wide, linear range of response ispreferred.

It is good engineering practice to develop the calibrationscheme at the beginning of the design process; too often,it is considered only after the system design is finalized.


74 Calibration and Measurement

Radiometric Calibration Philosophy

The most general principle is to calibrate a system in themanner it will be used. For example, if a distant pointsource is to be measured, a point-source irradiance cali-bration is the most desirable. However, depending on in-strument conditions and test equipment available, a goodresult might also be obtained by a radiance calibration cor-rected for the instrument’s solid angle. The following listdescribes some basic principles:

• Calibration requirements are driven by science andengineering goals.

• Make the calibration as independent of the specificinstrument as possible.

• Account for every factor influencing the calibration.• Select appropriate calibration standards.• Conduct an error assessment during the planning

phase to allow accurate estimation of data uncertain-ties.

• End-to-end calibration is preferred to summation ofindividual, component-level calibrations.

• Vary relevant external parameters (temperature,pressure, humidity, etc.) to determine their influenceon the calibration transfer function.

• Determine the calibration transfer function over theentire dynamic range of the instrument.

• Use more than one calibration configuration, if possi-ble, and compare results for consistency.

• Prior to the calibration of a flight instrument, conductthe entire procedure on a test dummy, prototype, orengineering model to uncover and fix any problemswith equipment or procedures.

• Inspect and interpret results early, while the instru-ment is still in the test position; this allows for imme-diate correction to equipment and procedures.

• Because calibration is often the last step on a GANTTchart prior to instrument delivery, it is most suscepti-ble to the “squeeze play.” Plan ahead!

• Adhere to the KISS (Keep It Simple, Smarty!) princi-ple as much as possible.



Distant Small Source Calibration

In a distant small source calibration, the inversesquare law applies. This is an irradiance calibration inwhich the relationship between output and input signalsis expressed as

RE =Signal× (d2/I)where

RE = irradiance responsivity [Signal/(W/m2)]d = source–system distance (m)I = source intensity (W/sr)

The primary advantage of this calibration method is thatalmost any source can be used, provided it is both “small”and “distant” enough to meet the inverse square law test.

There are several disadvantages, however:

• Signals are typically small.• There may be an intervening atmosphere.• Distance d must be known well (the error in RE is

twice the error in d due to the square term).• A background is present due to the fact that the

source image often does not fill the instrument’s fieldstop.

Examples of sources used in this method include black-body radiation simulators, small tungsten lamps used atsuitable distances, and the sun and stars.



Collimators and the Distant Small Source

Collimators—reflective and refractive—can improve thedistant small source calibration when a laboratory sourceis placed at the collimator focal point. The source imagemust, as above, be contained within the instrument’s fieldstop. An off-axis parabola is often used for this purpose. Inthis case, the irradiance responsivity is

RE =Signal× ( f 2/I)

where f is the collimator focal length.

Advantages:

• A controllable atmosphere (laboratory)• Controllable background• Possible use of nearly any small source• Larger signals than astronomical sources provide

Disadvantages:

• Must know the focal length f (typically a one-timemeasurement)

• Hardware can be expensive, both for collimator opticsand any chamber hardware (vacuum, for cooled, low-background infrared targets) in which they might beencased

A reflective on-axis collimator (centrally obscured systemsuch as a Cassegrain) may also be used. The responsivityequation is the same as above, as are the advantages.

Disadvantages:

• Presence of central obscuration• Diffraction possible from secondary mirror mount• A direct path into the collimator for source stray light• Difficulties in baffling



More on Collimators

The on-axis reflective system provides the advantage ofaccommodating instruments with wide fields of view. Aswith the off-axis parabola, advantages and disadvantageswill vary with instrument and application.

The refractive collimator provides an alternative to amirror-based system.

Advantages:

• Relatively simple alignment (in the visible region)due to an unfolded path

• Simple to set up, particularly if off-the-shelf compo-nents are used

Disadvantages:

• Wavelength range limited by τ(λ) of the material• Reflection losses due to refractive index• Ghost images due to reflections from optical elements• Wavelength dependence of anti-reflection coatings• Chromatic aberration from refractive optical ele-

ments• Possible difficulties in alignment if the lens is trans-

missive

The good news: many of these difficulties can be dis-missed or minimized if the collimator is used in thewavelength region for which it was optimized.



Extended Source Calibrations

For calibration purposes,extended sources areof two types, near anddistant. Radiance fromthe near extended sourcemust fill the instrument’saperture stop in additionto its field stop. In the distant extended source case, whichis often used by satellite or airborne instrumentation withrespect to a large, uniform ground target,atmospheric correction is required. The ra-diance responsivity for both cases is RL =Signal/L, where RL is in Signal/(W/m2sr).

Advantages:

• Distance d between source and radiometer is notimportant.

• There is no background (assuming proper test setup).• If the instrument is not precisely aligned (detector not

exactly at distance f from aperture), it doesn’t matter.• There is minimal intervening atmosphere (in the near

extended source case).

The presence of an intervening atmosphere is a disad-vantage in the distant extended-source case; and in bothcases, a large uniform source is required. Laboratorysources may include a blackbody radiation simulator, anintegrating sphere, or a transmissive or reflective diffuserwith a standard lamp. Uniform natural sources, such aslarge-area sites at White Sands, New Mexico, are amongthe distant extended sources used.

Radiative transfer programs used for atmospheric correc-tion include MODTRAN (son of LOWTRAN), FASCODE,and SMARTS in the US, and 5S and 6S in France.



Other Calibration Methods

The near small source (Jones source) calibrationmethod provides radiance responsivity and includesthe ratio of instrument aper-ture and source areas, RL =Signal/Ls × (Ao/As) with unitsin Signal/(W/m2 sr). The in-strument is focused at inifin-ity. Radiation from the smallsource placed within the conewill fill the instrument’s field stop, but not its aperture,hence the need for the ratio.

Advantages:

• Minimal atmosphere• Calibration source can be small

Disadvantage:

• Need to account for background radiation

This is a good method to use for some ground-basedastronomical telescopes, as the small source can bemounted to the side of the telescope and positioned infront of the aperture in calibration mode.

In the direct method, a power responsivity RΦ in Sig-nal/W is obtained by pointing a small, narrow beam ofknown flux at a radiometerwhose aperture and field stopswill be underfilled. A laserbeam is often used in thismethod.

Advantage:

• Simple setup

Disadvantages:

• Presence of background radiation• No accounting for aperture or field stop nonuniformi-

ties



The Measurement Equation

The measurement equation relates an instrument’soutput signal to input radiometric quantity (radiance,irradiance, flux) as a function of instrument responsivity.One version, in differential form, is

dSignal=R(λ)worldLλworlddAdΩdλ

where the usual suspects of spectral radiance, elementof area, and element of projected solid angle have beendiscussed at length; R(λ) is the instrument’s spectralresponsivity at a particular wavelength; and “world” refersto the environment surrounding the measurement, whichincludes:

θ, φ (angular dependence); x, y (position dependence); t, ν

(time, frequency); s, p (polarization components); alongwith diffraction, instrument and test setup nonlinearities,and “phase of the moon.”

In terms of an integral, the above equation looks like

Signal=Ñ

R(λ)LλdAapdΩdλ

where Lλ is input spectral radiance, related to sourcespectral radiance by the transmission of the interveningmedium. Other forms of the measurement equation existand may be formulated with respect to measurementconfiguration. For a point source, where Iλ is the pointsource’s spectral intensity and d is the source–aperturedistance,

Signal= 1/d2Ï

R(λ)IλdAapdλ



Errors in Measurements

Measurement errors are of two types: random and sys-tematic. Random errors are those that vary in an un-predictable manner when the same quantity is measuredrepeatedly under identical conditions. Precision is a termassociated with ran-dom error; a mea-surement is consid-ered precise if it isrepeatable. A nor-mal, Gaussian distri-bution of random er-ror is typically as-sumed. To reduce thevalue of random er-ror, increase the number of measurements and apply sta-tistical analysis to the result.

Systematic errors constitute a displacement between aquantity’s measured value and its true value. Accuracyis often used in discussion of systematic errors, many ofwhich arise from improper calibration and/or test setup.Unlike random errors, systematic errors are not revealedby repeated measurement.

In the third case above, the x-y average of the patternlies very close to the target center. Systematic andrandom errors must be considered together in evaluatinga measurement’s total uncertainty. Errors can bespecified as absolute, with the magnitude in appropriateengineering units, or fractionally, usually in percent.



Signal-to-Noise Ratio and Measurement Error

The fundamental limit to a measurement error is randomnoise. If all other error sources are reduced to zero, theremaining noise is most likely to be Gaussian, usually shotnoise or Johnson noise associated with the detector. Insuch a system, the measurement uncertainty is inverselyrelated to SNR: Measurement uncertainty = (1/SNR).

SNR (single measurement) Uncertainty1 1 (100%)10 0.1 (10%)100 0.01 (1%)1000 0.001 (0.1%)

Taking repeated measurements improves SNR. Theresultant SNR is proportional to the square root ofthe number of independent measurements, SNR ∝ p

N.Further analytical tools help with the dataset analysis.

Tool Formula

Mean x =∑i

xi

N, N = number of

measurements

Variance σ2 =∑i(xi − x)2

N −1

Standard deviation σ=p

σ2

Standard deviation of mean σx =σpN

An important contribution to random error (precision)is the granularity of the measurement instrument.Most electrical readings are now taken with digitalinstrumentation. In this case, quantization noise may becalculated as LSB = (Signalmax)/2n, where Signalmax isthe full-scale signal, LSB is the magnitude of the leastsignificant bit, and n is the number of bits.



The Range Equation

The range equation provides the distance at whicha specified target will generate a particular SNR ratiofrom a passive electro-optical system whose parametersare specified. Its components can be separated into fourcategories:

R = (Isτa)1/2 ×[π

4Doτo

( f /#)pΩ

]1/2× (D∗)1/2 ×

(1

SNRp

B

)1/2

1 2 3 4

Category Includes1: Target Source intensity, intervening

atmosphere transmission2: Optical system Receiver diameter Do, f /#,

solid angle FOV, opticstransmission

3: Detector D∗4: Signal processing Required SNR and noise

bandwidth

The range equation can be permuted to calculate theexpected SNR for a specified target range.

Several other equations exist for assessing the perfor-mance of imaging and non-imaging sytems as a function ofseveral parameters. For example, the noise equivalentirradiance (NEI) for a target that underfills an instru-ment’s FOV is

NEI = 4( f /#)Ω1/2B1/2

πDoD∗τo



Radiometric Temperatures

A number of temperature terms are used in radiometryand photometry; the definitions below refer to an objectwhose temperature is unknown. The object’s radiationtemperature is the temperature of a blackbody havingthe same total radiance as the unknown. Its radiancetemperature (brightness temperature) is the temper-ature of a blackbody that has the same radiance in a nar-row spectral band as the unknown.

The object’s ratio temperature is the temperature of ablackbody with the sameratio of radiances at twowavelengths as the un-known:Lλ1BB

Lλ2BB= Lλ1unknown

Lλ2unknown

The distribution tem-perature is the temper-

ature of a blackbody whose radiance values, over a signifi-cant portion of the spectrum, are most nearly proportionalto those of the unknown. Mathematically, it is the temper-ature of the blackbody for which the following integral isminimized by successive selections of the constant a andtemperature T (M′

λ is the measured test source exitance):∫ λ2

λ1

[1− (M′λ/aMλ(T))]2dλ

An object’s color temperature is the temperature of ablackbody having the same chromaticity (color) as the un-known. In many applications, correlated color temper-ature (CCT), the temperature of a blackbody having themost similar chromaticity, is used instead.

Sun 5750 K λpeak = 0.5 µmTungsten lamp 3000 K λp = 1 µmEarth 300 K λp = 10 µm


Photometry 85

Photometric Quantities

Photometry is a subset of radiometry in which theinput optical stimulus is modified by the response of thehuman eye. Its basic quantities are defined similarly toradiometric quantities, with subscript v (for visual) added.


Luminousenergy

Qv — lm ·s

Luminouspower

ΦvdQv

dtlm

Luminousintensity

IvdΦv

dωlm/sr (cd)

Luminousexitance

MvdΦv

dAlm/m2

Illuminance EvdΦv

dAlm/m2

Luminance Lvd2Φv

dAdωcosθlm/m2sr (cd/m2)

Lumens/m2 (lm/m2) are also called lux (lx); lm ·s (a unit ofenergy) is also called talbot.

The candela (cd) is one of seven SI base quantities fromwhich other quantities are derived: “The candela is theluminous intensity, in a given direction, of a source thatemits monochromatic radiation of frequency 540×1012 Hzand that has a radiant intensity in that direction of1/638 W sr−1” [see Ref. 3]. Note that the wavelengthcorresponding to the given frequency is 555 nm.


86 Photometry

Human Visual Response

The dynamic range of human vision is about 11 orders ofmagnitude, but the eye’s spectral response is confined toa narrow region of the electromagnetic spectrum. Two im-portant curves (“efficiency functions”) defined by the CIErepresent the “light-” and “dark-adapted” eye functions—the photopic and scotopic curves, respectively.

Each curve features a response by one of the two typesof visual photoreceptors, rods and cones—the cones re-sponding to photopic conditions and the rods to scotopic.The “in-between” state of illumination, where both rodsand cones respond, is called mesopic.

In order to solve problems in photometry with dataacquired from radiometric instruments, the radiometricdistribution, for example, power, must be folded into anintegral with the V (λ) function over the photopic limits (forlight adaptation):

Φv = Km

∫ 760

380V (λ)Φλdλ

Similar formulations apply for Mv, Iv, etc. The constantKm is called the luminous efficacy; its value is 683 lm/Wat 555 nm. For the scotopic curve, the formulation issimilar with Φ′

v replacing Φv, and V ′(λ) replacing V (λ). Theconstant is K ′

m, whose value is 1700 lm/W at 507 nm. Theunits of both Φv and Φ′

v are lumens.


Photometry 87

Color

The eye’s cones provide the capability for color vision,with three curves called color matching functions thatdescribe the response of the CIE Standard Observer.The 1931 CIE Standard Observer employs a 2-deg FOV,with y(λ)=V (λ).

The three curves allow calculation of the tristimulusvalues X , Y , and Z that completely describe a test colorin terms of the three primaries—red, green, and blue:

X = Km

∫ 760

380Φλρ(λ)x(λ)dλ Y = Km

∫ 760

380Φλρ(λ)y(λ)dλ

Z = Km

∫ 760

380Φλρ(λ)z(λ)dλ

Φλ is incident radiant power, ρ(λ) is spectral reflectance,and Km = 683 lm/W.

Normalizing these values by(X + Y + Z) gives the chro-maticity coordinates x, y,and z (z = 1−x−y), which spec-ify any color in 2D space viathe 1931 CIE chromaticity di-agram, whose general shapeis indicated here.


88 Photometry

Sources and the Eye’s Response

It is instructive to consider the spectral distribution fromthe product of a source function and the eye’s response tolight. Data taken with a USB spectrometer and plottedat 5-nm increments, with response normalized to 1 at555 nm, show the peaks from a typical fluorescent officelight or compact fluorescent light (CFL) bulb againstthe smooth function of a laboratory tungsten–halogenincandescent source.

When the photopic curve is applied to the data, the peakat 545 nm stands out as the most prominent fluorescentfeature, the larger peak at 435 nm having been minimizedby the eye’s response. Before choosing a source, be sure tounderstand its response on the chosen detector.


Appendices 89

SI Base Quantities, Prefixes, and Uncertainty Reporting

SI Base Quantities

Quantity Name Symbollength meter mmass kilogram kgtime second selectric current ampere Athermodynamic temperature kelvin Kamount of substance mole molluminous intensity candela cd

SI Prefixes

Factor Prefix Symbol Factor Prefix Symbol1018 exa E 10−1 deci d1015 peta P 10−2 centi c1012 tera T 10−3 milli m109 giga G 10−6 micro µ

106 mega M 10−9 nano n103 kilo k 10−12 pico p102 hecto h 10−15 femto f101 deka da 10−18 atto a

Concise Notation and Uncertainty Reporting

The absolute uncertainty of a measurement is its es-timated standard deviation σ, such that, for an approx-imately Gaussian distribution of error, 68% of the mea-sured value falls within the limits of the value +/−1σ.For example, the absolute uncertainty in the value of c2,0.014387752 m K, is 0.000000025 m K; the expression be-comes 1.4387752×10−2+/−0.000000025 m K. The constantand its standard deviation may also be expressed in con-cise notation as 1.4387752(25)×10−2 m K, where the (25)refers to the corresponding last digits of the constant.

A relative uncertainty may be expressed. This is theabsolute uncertainty divided by the value of the constant.In the case of c2, it is 1.7×10−6. The entire expression isthen c2 = 1.4387752(25)×10−2 m K (1.7×10−6).


90 Appendices

Physical Constants: 2010 CODATA Recommended Values

Quantity Symbol Value Units Relativeuncertainty

Speed oflight(vacuum)

c, c0 299,792,458 m/s exact

Magneticconstant

µ0 4π×10−7 N/A2 exact

Electricconstant(1/µ0c2)

ε0 8.854187 . . .×10−12

F/m exact

Planckconstant

h 6.62606957(29)×10−34

J s 4.4×10−8

Electroniccharge

q, e 1.602176565(35)×10−19

C 2.2×10−8

Boltzmannconstant

k 1.3806488(13)×10−23

J/K 9.1×10−7

Boltzmannconstant

k 8.6173324(78)×10−5

eV/K 9.1×10−7

Stefan–Boltzmannconstant

σ 5.670373(21)×10−8

W/(m2K4) 3.6×10−6

Firstradiationconstant(2πhc2)

c1 3.74177153(17)×10−16

W m2 4.4×10−8

Firstradiationconstantfor Lλ

c1L 1.191042869(53)×10−16

Wm2/sr 4.4×10−8

Secondradiationconstant(hc/k)

c2 1.4387770(13)×10−2

m K 9.1×10−7

Wiendisplacementlaw constant

b 2.8977721(26)×10−3

m K 9.1×10−7


Appendices 91

Source Luminance Values

Astronomical Sources

Source Luminance (cd/m2)Night sky, cloudy, no moon 10−4

Darkest sky 4×10−4

Night sky, clear, no moon 10−3

Night sky, full moon 10−2

Clear sky 0.5 h after sunset 0.1Clear sky 0.25 h after sunset 1Cloudy sky at sunset 10Gray sky at noon 102

Cloudy sky at noon 103

Moon 2.5×103

Average clear sky 8×103

Clear sky at noon 104

Solar disc 1.6×109

Lightning 8×1010

Practical Sources

Source Luminance (cd/m2)Minimum of human vision 3×10−6

Scotopic vision valid <0.003Photopic vision valid >3Green electroluminescent 25T8 cool white fluorescent 104

Acetylene burner 105

60 W inside frosted lamp 1.2×105

Candle 6×105

Sodium vapor lamp 7×105

High-pressure Hg vapor lamp 1.5×106

Tungsten lamp filament 8×106

Plain carbon arc crater 1.6×108

Cored carbon arc crater 109

Atomic fusion bomb 1012


92 Appendices

More Source Values

Illuminance of Various Sources

Source Illuminance (lm/m2 = lux)Absolute minimum forvisual magnitude, Mv = 8

1.6×10−9

Typical minimum, Mv = 6 10−8

0-magnitude star outsidethe atmosphere

2.54×10−6

Venus 1.3×10−4

Full moonlight 1Street lighting (typical) 10Recommended reading 102

Workplace lighting 102–103

Lighting for surgery 104

Luminous Intensity Values

Source Luminous intensity (cd)Bicycle headlamp withoutreflector, any direction

2.5

Bicycle headlamp withreflector, center of beam

250

Incandescent reflectorlamp, PAR38E Spot,120 W, center of beam

104

Lighthouse, center of beam 2×106

Astronomical Conversion Factors

1 Jansky (Jy) 10−26 W/m2 Hz(spectral irradiance)

1 W/cm2 ·µm 3×1016/λ2 Jy1 Rayleigh 106 photons/cm2 s1 S10 (number of 10th

magnitude stars persquare degree)

1.23×10−12 W/cm2 sr µm at0.55 µm

1 S10 1.899×106 photons/s cm2 sr µmat 0.55 µm


Appendices 93

Solid Angle Relationships

Θ1/2(deg)

Θ1/2(rad)

ω (sr) Ω (sr) ω/Ω f /#

0.573 .0100 .00031 .00031 1.000 50.001.000 .0175 .00096 .00096 1.000 28.651.146 .0200 .00126 .00126 1.000 25.001.719 .0300 .00283 .00283 1.000 16.672.000 .0349 .00383 .00383 1.000 14.332.292 .0400 .00503 .00502 1.000 12.502.865 .0500 .00785 .00785 1.001 10.003.000 .0524 .00861 .00861 1.001 9.5544.000 .0698 .0153 .0153 1.001 7.1685.000 .0873 .0239 .0239 1.002 5.7375.730 .1000 .0314 .0313 1.003 5.00810.00 .1745 .0955 .0947 1.008 2.88011.46 .2000 .1252 .1240 1.010 2.51715.00 .2618 .2141 .2104 1.017 1.93217.19 .3000 .2806 .2744 1.023 1.69220.00 .3491 .3789 .3675 1.031 1.46222.92 .4000 .4960 .4764 1.041 1.28425.00 .4363 .5887 .5611 1.049 1.18328.65 .5000 .7692 .7221 1.065 1.04330.00 .5236 .8418 .7854 1.072 1.00034.38 .6000 1.097 1.002 1.096 0.88640.11 .7000 1.478 1.304 1.133 0.77645.00 .7854 1.840 1.571 1.172 0.70745.84 .8000 1.906 1.617 1.179 0.69751.57 .9000 2.377 1.928 1.233 0.63857.30 1.000 2.888 2.224 1.298 0.59460.00 1.047 3.142 2.356 1.333 0.57771.63 1.250 4.269 2.830 1.521 0.52785.95 1.500 5.839 3.126 1.868 0.50190.00 1.571 6.283 3.142 2.000 0.500(Adapted from Ref. 4)

The percentage error using ω rather than Ω is less than5% when Θ1/2 is 25 deg; less than 2% when Θ1/2 is 16 deg;and less than 1% when Θ1/2 is 10 deg.


94 Appendices

Rays, Stops, and Pupils

The chief ray inan optical systemoriginates at theedge of the object,passes through thecenter of the en-trance pupil (NP), the center of the aperture stop (AS),the edge of the field stop (FS), and the center of the exitpupil (XP). The chief ray defines image height and trans-verse (lateral) magnification. There may be several inter-mediate pupil planes in a complex optical system.

The marginal (rim)ray originates at theintersection of the ob-ject and optical axis.It passes through theedge of the entrance

pupil, touches the edge (rim) of the aperture stop, and pro-ceeds through the center of the field stop to the edge of theexit pupil. The marginal ray defines the system’s longitu-dinal magnification and location of the image.

The entrance pupil is the image of the aperture stop inobject space (as seen from the object), while the exit pupilis the image of the aperture stop in image space (as seenfrom the image).


Appendices 95

Diffraction

In imaging applications, a point source such as a stardoes not image exactly to a point but creates a “blur”on the instrument focal plane. The blur arises due tothe non-ideal nature of optical instrument characteristics,which result in rays being diffracted (bent). The effect ismost pronounced for small, widely spaced apertures andlonger wavelengths; hence, it can challenge designers of IRspectrometers, for example. Diffraction is less severe forshorter wavelengths and large, closely spaced apertures.

An optical system free (at least in theory) of opticalaberrations is said to exhibit diffraction-limitedperformance, where the spreading of the image on thefocal plane is due to diffraction effects alone.

The diffraction patternon the focal plane takesthe form of the Airydisk which, as imagedfor a circular aperture,appears as a series of

concentric rings, with 84% of the energy in the imagecontained in the central circle. The radius of the circle iscalculated as 1.22 λ ( f /#), with f /# defined previously, andλ the wavelength of incoming optical radiation (or centralwavelength of the incoming beam; the narrower, the moreaccurate).

Two “adjacent” objects are considered resolved, accordingto the Rayleigh criterion, whenthe diffraction pattern peak ofobject #1 falls on the first minimum(first dark ring) of object #2.

Diffraction is typically studied indepth as part of physical optics.


96 Appendices

Action Spectra and Optical Radiation Regions

An action spectrum is a biological spectral response toan input optical stimulus. For example, action spectradescribe the blue light hazard between about 400–550 nmencountered by welders, and the erythemal action (skinreddening) response to UV radiation from the sun andtanning lamps.

Mathematically, the action spectrum is handled as an in-strument responsivity, placed in an integral along withthe input source functionand integrated over thecommon spectral band.Depending on the phe-nomenon measured, mul-tiplicative constants mayalso be applied to the re-sult.

UV Band Designations—Three Separate Authorities(all wavelengths in nm)

Band CIE NOAA ANSIUVC 100–280 100–280 200–280UVB 280–315 280–320 280–315UVA 315–400 320–400 315–400

The UVA region is further subdivided into UVA1 andUVA2 due to differing responses to optical radiation,with UVA2 comprising 315(320) nm to 340 nm, andUVA1 extending from 340 nm to 400 nm. The erythemalaction spectrum is most pronounced in the UVB region.“Facultative pigmentation,” i.e., tanning, is the result ofoptical radiation in the UVA1.

Phototherapy in the UVA1 region of the spectrum iscurrently used in treatments for lupus and many skindiseases.


97

Equation Summary

Optical terms and basic equations

n = c/v λ0 = nλ

Snell’s law:

n1 sinθ1 = n2 sinθ2

Numerical aperture:

NA= nsinΘ1/2

f /# for object at infinity:

f /#= fD

f /# (preferred definition):

f /#= 12sinΘ1/2

Working f /#:

f /#′ = ( f /#)[1+ (m/mp)

]Projected area, solid angle, projected solid angle

Projected area:

Ap =∫

AcosθdA

Solid angle:

ω=∫φ

∫θ

sinθdθdφ

Solid angle of a right circular cone:

ω= 2π(1−cosΘ1/2)

Projected solid angle of a right circular cone:

Ω=πsin2Θ1/2 = π

4( f /#)2


98

Equation Summary

Radiometric terminology and propagation laws

Radiant power:

Φ= dQdt

Radiant intensity:

I = dΦdω

Radiant exitance:

M = dΦdA

Irradiance:

E = dΦdA

Radiance:

L = d2Φ

dAdωcosθ

Photon flux at wavelength λ0:

Φq =Φλ0

hc

Relationship between intensity, radiance and pro-jected area:

I = LAp

Lambert’s law:

I = I0 cosθ

Lambertian approximation:

L(θ,φ)=Constant


99

Equation Summary

Lambertian surface relationships:

M =πL L = Eρπ

Inverse square law of irradiance:

E = I cosθd2

Cosine3 law of irradiance:

E = I cos3θ

d2

Cosine4 law of irradiance:

E = LAs cos4θ

d2

Throughput:

T = AΩ

Basic throughput:

n2T = n2 AΩ

Equation of radiative transfer, differential form:

d2Φ1→2 = L(θ,φ)dA1 cosθ1dA2 cosθ2

d2

Equation of radiative transfer, integral form:

Φ1→2 =∫

A2

∫A1

L(θ,φ)cosθ1 cosθ2dA1dA2

d2

Configuration factor:

F = Φ1→2

Φ1


100

Equation Summary

Radiant power gain when Alens is the first surface ina system, and Adet is the second surface:

G = τlens Alens

Adet

Camera equation:

E t = πLs

4( f /#)2(1+m)2

Extended source irradiance at focal plane:

E t = τoτatmπ fv(1− A2)Ls(θ,φ)cosnθ

4( f /#)2(1+m)2

Point source irradiance at focal plane:

E t = τoτatm fv(1− A2)Is cosnθ

d2

Simplification of the equation of radiative transfer:

Φ= LAΩ

Resulting irradiance on area A:

E =Φ/A

Material properties

Total transmittance:

τ= Φt

Φi=

∫ ∞0 τ(λ)Φλidλ∫ ∞

0 Φλidλ

Total reflectance:

ρ= Φr

Φi=

∫ ∞0 ρ(λ)Φλidλ∫ ∞

0 Φλidλ


101

Equation Summary

Total absorptance:

α= Φa

Φi=

∫ ∞0 α(λ)Φλidλ∫ ∞

0 Φλidλ

Total emittance:

ε=Ðε(θ,φ;λ)LλBB sinθcosθdθdφdλ

(σ/π)T4

Spectral emittance:

ε(λ)= Lλ(source)/LλBB

BRDF:

fr(θi,φi;θr,φr)= dLr(θr,φr)dE i(θi,φi)

= dLr(θr,φr)L i(θi,φi)dΩi

sr−1

Conservation of energy in a beam:

τ+ρ+α= 1

Kirchoff’s law:

α= ε

Single-surface reflectivity and transmissivity, nor-mal incidence and no absorption:

τss = 4n1n2

(n2 +n1)2; ρss =

(n2 −n1

n2 +n1

)2

Internal transmittance at wavelength λ:

τi(λ)= e−α′(λ)x

Optical thickness (optical depth) at wavelength λ:

τo(λ)=α′(λ)x


102

Equation Summary

Transmissivity, infinite number of passes:

τ∞ = (1−ρss2)τi

1−ρss2τi

2

Reflectivity, infinite number of passes:

ρ∞ = ρss + ρss(1−ρss)2τi2

1−ρss2τi

2

Absorptivity, infinite number of passes:

α∞ = (1−ρss)(1−τi)1−ρssτi

Generation of optical radiation

Planck’s law:

Lλ = c1

πn2λ51

ec2/nλT −1

Stefan–Boltzmann law:

M =σT4

Wien displacement law:

λmaxT = 2898 µm K

Wien approximation:

Lλ = c1

πλ5 e−c2λT

Rayleigh–Jeans law:

Lλ = 2ckT

λ4

Lambert–Bouguer–Beer law:

Eλ = Eoλe−τo(λ)m


103

Equation Summary

Detectors of optical radiation

Responsivity:

R=∫

R(λ)Φλdλ∫Φλdλ

Signal-to-noise ratio:

SNR= vs

vn= is

in

Impedance:

Z = dVdI

Detectivity:

D = 1NEP

= SNRΦinput

Noise equivalent power:

NEP= vn

R=Φvn

vs

Specific detectivity:

D∗ = D√

AdB =√

AdB

NEP=

√AdB

Φ

vs

vn

Specific detectivity for photons at wavelength λ:

D∗q = D∗ hc

λ

D∗BLIP for a photoconductive detector:

D∗BLIP(λ, f )= λ/2hc× (η/Eq)1/2

D∗BLIP for a photovoltaic detector:

D∗BLIP(λ, f )=

√ηλ

2Ebackhc


104

Equation Summary

Cutoff wavelength in micrometers:

λc(µm)= 1.24Eg(eV )

= 1.24φ(eV )

Photoconductive gain:

G = τl

τtr

Photodiode current in a photovoltaic device in thepresence of incident radiation:

I = I0

(e

qVβkT −1

)− Ig

Detector noise equations

Effective noise bandwidth:

B = 1G( f0)

∫ ∞

0G( f )d f

Mean-square noise voltage:

v2 = (vi −vavg)2 = 1T

∫ T

0(vi −vavg)2dt

Johnson noise:

v j =p

4kTRB

Johnson noise when B is established by an RCcircuit time constant:

v j =√

kT/C

Shot noise:

is =√

2qIdcB


105

Equation Summary

Generation–recombination noise:

vG−R = 2iR

√τlB

N[1+ (2π f τl)2

]1/f noise:

i1/ f =√

KsIαdcB

f β

Photon noise: √(∆n)2 =

√n

Temperature fluctuation noise:

∆T =√

4kT2KB

K2 +4π2 f 2H2

Quantization noise:

LSB= SIGNALmax

2n

Spectrometry

Rayleigh resolution criterion:

Resolution= 1.22λ( f /#)

Resolving power of a prism-based instrument:

Qrp = bdndλ

Throughput of a prism-based instrument:

T = Aslit Aprism cosθ/ f 2

Grating equation:

mλ= d(sinθi +sinθd)


106

Equation Summary

Resolving power of a grating-based instrument:

Qrp = mN

Throughput of a grating-based instrument:

T = (Aslit Agrating cosθ)/ f 2

Neutral density filter number:

n =− log(Efiltered/Eincident)

Radiometric calibration

Irradiance responsivity, distant small source:

RE =Signal× (d2/I)

Irradiance responsivity for a distant small sourceused with a collimator of focal length f :

RE =Signal× ( f 2/I)

Radiance responsivity for an extended sourcecalibration:

RL =Signal/L

Radiance responsivity for a near small source(Jones source) calibration:

RL =Signal/Ls × (Ao/As)

Power responsivity for direct calibration method:

RΦ =Signal/Φ


107

Equation Summary

Measurement and range equations, NEI, and tempe-ratures

Measurement equation for an extended source:

Signal=Ñ

R(λ)LλdAapdΩdλ

Measurement equation for a point source:

Signal= 1/d2Ï

R(λ)IλdAapdλ

Range equation:

R = (Isτa)1/2[π

4Doτo

( f /#)pΩ

]1/2(D∗)1/2

(1

SNRp

B

)1/2

Noise equivalent irradiance:

NEI = 4( f /#)Ω1/2B1/2

πDoD∗τo

Ratio temperature:

Lλ1BB

Lλ2BB= Lλ1unknown

Lλ2unknown

Distribution temperature:∫ λ2

λ1

1− (M′λ/aMλ(T))2 dλ

Photometry

The V(λ) function:

Φv = Km

∫ 760

380V (λ)Φλdλ

The V′(λ) function:

Φ′v = K ′

m

∫ 760

380V ′(λ)Φλdλ


108

Equation Summary

Tristimulus values:

X = Km

∫ 760

380Φλρ(λ)x(λ)dλ Y = Km

∫ 760

380Φλρ(λ)y(λ)dλ

Z = Km

∫ 760

380Φλρ(λ)z(λ)dλ


109

References

1. P. N. Slater, Remote Sensing: Optics and OpticalSystems, p 119, Addison–Wesley, Reading, MA, 1980.

2. F. Grum and R. J. Becherer, Radiometry, Vol. 1 inOptical Radiation Measurements series, F. Grum, Ed.,p 115, Academic Press, New York (1979).

3. Comptes Rendus de la 16e CGPM (16th ConférenceGénérale des Poids et Mesures), Resolution 3, 1979.http://www.bipm.org/en/CGPM/db/16/3/.

4. F. E. Nicodemus, Ed., Self Study Manual on OpticalRadiation Measurements, Part I—Concepts, NBSTechnical Note 910-1, pp 99–100, US National Bureauof Standards, Washington, DC, 1976.


110

Bibliography

Boyd, R. W., Radiometry and the Detection of OpticalRadiation, John Wiley and Sons, New York (1983).

Budde, W., Physical Detectors of Optical Radiation,Optical Radiation Measurements, v. 4, Grum, F. andBartleson, C. J., Eds., Academic Press, New York (1983).

Dereniak, E. L. and Crowe, D. G., Optical RadiationDetectors, John Wiley & Sons, New York (1984).

Dereniak, E. L. and Dereniak, T. D., Geometrical andTrigonometric Optics, Cambridge University Press, Cam-bridge, UK (2008).

Greivenkamp, J. E., Field Guide to Geometrical Optics,SPIE Press, Bellingham, WA (2004)[doi:10.1117/3.547461].

Holst, G. C., Common Sense Approach to Thermal Imag-ing, JCD Publishing, Winter Park, FL, and SPIE Press,Bellingham, WA (2000).

Howell, J. R., A Catalog of Radiation Heat TransferConfiguration Factors.http://www.engr.uky.edu/rtl/Catalog/intro.html.

NIST Physical Measurement Laboratory, CODATA Inter-nationally Recommended Values of the Fundamental Phys-ical Constants, 2006.http://physics.nist.gov/cuu/Constants/index.html.

Palmer, J. M. and Grant, B. G., The Art of Radiometry,SPIE Press, Bellingham, WA (2009)[doi:10.1117/3.798237].

Refractive Index.Info. http://refractiveindex.info/.

Robertson, A., “R2-35 Uncertainties in DistributionTemperature Determination,” Report to CIE Division 2,Ottawa, Canada, 2005.

Vincent, J. D., Fundamentals of Infrared Detector Opera-tion and Testing, John Wiley & Sons, New York (1990).


111

Bibliography

Wolfe, W. L. and Zissis, G. J., Eds., The Infrared Handbook,US Government, Washington, DC (1978), and SPIE Press,Bellingham, WA (1985).

Wyatt, C. L., Electro-optical System Design for InformationProcessing, McGraw Hill, New York (1991).

Wyatt, C. L., Radiometric Calibration: Theory andMethods, Academic, New York (1978).


112

Index

3-dB frequency, 49α/ε, 31AΩ product, 15absorptance

spectral, 30total, 30

absorption, 27, 30absorptivity, 35accuracy, 81action spectrum, 96airmass, 43Airy disk, 95angle

plane, 3solid, 3

aperture stop (AS), 21, 94area-solid pair, 16artificial sources

luminescent, 45thermal, 45

ASTM G-173-03 referencespectrum, 43

background-limitedinfrared photodetector(BLIP), 50

baffle, 69bidirectional reflectance

distribution function(BRDF), 29

bidirectionaltransmittancedistribution function(BTDF), 28

blackbody radiation, 38bolometer, 57calibration

absolute, 73distant small

source, 75relative, 73

camera equation, 26charge transfer efficiency

(CTE), 54charge-coupled device

(CCD), 66charge-injection device

(CID), 66chief ray, 94choppers, 68chopping factor, 68chromaticity coordinates,

87CIE Standard Observer,

87collimator, 76

refractive, 77color matching functions,

87color temperature, 84compact fluorescent lamp

(CFL), 46complementary

metal-oxidesemiconductor(CMOS), 66

concise notation, 89conduction band, 59conductivity, 60cone, 86configuration factors, 19conservation of energy, 30correlated color

temperature (CCT),84

cosine correctors, 69cosine3 law of irradiance,

14cosine4 law of irradiance,

14


113

Index

Curie temperature, 49, 58cutoff frequency

thermal, 49cutoff wavelength, 48, 59,

61depletion region, 62derived quantities, 6detector

area, 48photon, 48point, 48thermal, 48

diffraction, 95diffraction-limited

performance, 95direct method, 79distribution temperature,

84effective noise bandwidth

(ENB), 50emission, 31emittance

spectral, 31total, 31

entrance pupil (NP), 94equation of radiative

transfer, 18exit pupil (XP), 94exitance

radiant, 9, 10exposure

radiant, 6extended source

far, 78near, 78

external photoeffect, 61extrinsic semiconductors,

59

f /#working, 5

Fermi level, 59field lens, 22field stop (FS), 21, 94figures of merit, 51

D∗∗, 51detective

quantumefficiency, 51

detectivity, 51noise equivalent

power (NEP),51

specificdetectivity, 51

filterabsorption, 72neutral density

(ND), 72thin film

interference, 72fluorescent sources, 46Fresnel equations, 32gap energy, 59grating, 70, 71grating equation, 71graybody, 38image plane irradiance,

26impedance, 50incandescent, 46incoherent, 18instantaneous field of

view (IFOV), 65intensity, 11

radiant, 11intrinsic semiconductors,

59


114

Index

invarianceradiance, 17

inverse square law ofirradiance, 13

irradiance, 9Kirchoff ’s law, 42Lambert–Bouguer–Beer

law, 34, 44Lambertian disk, 23Lambertian source, 12Lambert’s law, 12Langley plot, 44laser, 47light-emitting diode

(LED), 47luminous efficacy, 86magnification

pupil, 5marginal (rim) ray, 94mean-square noise

voltage, 52measurement equation,

80mesopic, 86monochromator, 70n-type, 62natural sources

active, 43passive, 43

near small source (Jonessource), 79

noise, 50, 521/ f , 53amplifier, 54CCD, 54factor, 52figure, 52

Generation-Recombination(G-R), 53

Johnson, 53kT/C, 54microphonic, 54photon, 54power, 52quantization, 54shot, 53temperature, 52temperature

fluctuation, 53triboelectric, 54

noise equivalentirradiance (NEI), 83

numerical aperture (NA),5

Ohm’s law, 52optical depth, 44optical thickness, 34

spectral, 44order number, 71organic light-emitting

diode (OLED), 47p-n junction, 62p-type, 62Peltier coefficient, 56Peltier effect, 56photocathode, 61photoconductive gain, 60photomultiplier tube

(PMT), 61photon detection, 59photon quantities, 7photopic curve, 86photovoltaic detector, 62physical optics, 95


115

Index

pixel, 65Planck’s law, 38point source, 11

isotropic, 12potential difference, 62precision, 81prism, 70, 71projected area, 4projected solid angle, 4pyroelectric coefficient, 58pyroelectric effect, 58RA product, 50radiance, 9, 10

basic, 17temperature, 84

radiationionizing, 1

radiation referencesource, 68

radiation temperature, 84radiometry, 2random errors, 81range equation, 83ratio temperature, 84Rayleigh criterion, 70, 95Rayleigh-Jeans law, 40reflectance

spectral, 29factor, 30total, 29

reflection, 27, 29reflectivity, 35

single-surface, 33,35

specular, 32total, 33

refractive index, 2resolving power, 70

responsive quantumefficiency, 49

responsivity, 48irradiance, 49, 75photon, 48radiance, 49, 78

reverse saturationcurrent, 63

rod, 86root-mean-square (rms)

value, 49scotopic curve, 86Seebeck coefficient, 56selective radiator, 38signal, 50signal-to-noise (SNR)

ratio, 50Snell’s law, 2solar diffusing panel, 68spectral absorption

coefficient, 34spectral extinction

coefficient, 34spectral instrument, 70spectral radiant

quantities, 8spectrometer, 70spectroradiometer, 70sphere

integrating, 25Lambertian, 24

Stefan–Boltzmann law, 39systematic errors, 81targets, 36temperature coefficient of

resistance (TCR), 57thermal conductance, 55thermal detection, 55


116

Index

thermal resistance, 55thermal time constant, 55thermalization loss, 59thermoelectric effect, 56thermopile, 56throughput

basic, 15time constant

thermal, 49transit time, 60transmission, 27, 28transmissivity, 35

single-surface, 33,35

specular, 32total, 33

transmittance, 44internal, 34, 35spectral, 28total, 28

tristimulus values, 87uncertainty

absolute, 89relative, 89

valence band, 59velocity of light, 2velocity of propagation, 2Wien approximation, 40Wien displacement law

(Wien’s law), 39, 41work function, 61


Barbara G. Grant received her B.A. inmathematics from San Jose State Uni-versity in 1983, and her M.S. in opticalsciences from the University of Arizonain 1989, where her graduate researchfocused on the absolute radiometric cal-ibration of spaceborne imaging sensors.She was employed in the aerospace anddefense industry prior to attending the

University of Arizona and afterward, working at com-panies such as Lockheed Missiles and Space and NASAcontractors. She received two NASA Group AchievementAwards for her work on the GOES imager and sounder.

In 1994 Grant began her consultancy, Lines and LightsTechnology, which provides consulting and training indiverse applications of electro-optical technology includinginfrared sensors, UV measurement, and systems engineer-ing/systems analysis. Barbara enjoys teaching and hastaught several short courses for SPIE and in other venues.She has been interviewed on technical subjects for printand Internet media, and has appeared on radio andtelevision. In December 2009, she completed The Art ofRadiometry (SPIE Press) as co-author to the late JimPalmer, who had mentored her in the subject for manyyears.

SPIE Field Guides The aim of each SPIE Field Guide is to distill a major fi eld of optical science or technology into a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena.

Written for you—the practicing engineer or scientist—each fi eld guide includes the key defi nitions, equations, illustrations, application examples, design considerations, methods, and tips that you need in the lab and in the fi eld.

John E. GreivenkampSeries Editor

P.O. Box 10Bellingham, WA 98227-0010ISBN: 9780819488275SPIE Vol. No.: FG23

RadiometryBarbara G. Grant

Based on the SPIE bestseller, The Art of Radiometry by James M. Palmer and Barbara G. Grant, this Field Guide provides a practical, hands-on approach to the subject that the engineer, scientist, or student can use in real time. Written from a systems engineering perspective, this book covers topics in optical radiation propagation, material properties, sources, detectors, system components, measurement, calibration, and photometry. Appendices provide material on SI units, conversion factors, source luminance data, and many other subjects. The book’s organization and extensive collection of diagrams, tables and graphs will enable the reader to effi ciently identify and apply relevant information to radiometric problems arising amid the demands of today’s fast-paced technical environment.

www.spie.org/press/fi eldguides

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